Does "Heaven" have the same meaning as "God" here?
Couldn't keep it in,
Heaven knows I tried.
It is in the animation "Frozen" in the Elsa thought.
Just another question: do we have to write "Heaven" or "heaven" when it is not the first word of the sentence?
I should make a couple of remarks.
Do we have to write Heaven or heaven when it is not first in the sentence?
It depends. If it is a reference to, for example, the Christian one, it appears that you can choose to capitalize it. Even the Christian bible has an instance in which it is not capitalized. As for the reference to the Chicago Manual of Style
8.108 Heaven, hell, and so on
Terms for divine dwelling places, ideal states, places of divine punishment, and the like are usually lowercased (though they are often capitalized in a purely religious context)
So if your audience understands that you are writing in a religious context, then you are free to capitalize it.
If it is a generic or secular heaven and not a proper noun, then it should be written as heaven.
Returning to the lyrics, I understood them to be
Couldn't keep it in
Heaven knows I tried
with the line break and without the period. While writing a post, people often hit enter/return for a line break, but the line break doesn't display. So, I assumed that's what OP meant to display. It makes sense that there are no punctuation marks since lyrics found online are generally poorly written and often omit proper punctuation. Having it displayed this way, "Heaven" is at the beginning and should be capitalized.
If the word "H/heaven" is found elsewhere in the sentence, like at this site
Couldn't keep it in, heaven knows I tried!
then I believe it should not be capitalized, for the reason I give below.
Does "Heaven" have the same meaning as god here?
I assume you meant God (for example, in the Christian sense). Since this is a Disney movie and Disney does not produce religiously affiliated material, heaven would be a secular term and not a proper noun. Since we are speaking of heaven in a secular sense, then no, heaven does not mean the same thing as God. However, the expression is still understood to have the same meaning as "God knows".
Now, if it was, for example, a Christian person writing and chose to write Heaven instead of heaven, then yes, it is possible that Heaven meant the same thing as God. Heaven is often a euphemism or allusion to God in Christianity. So one could say that, in such an instance, Heaven meant the same thing as God. Further, "Heaven knows" would also mean the same thing as "God knows".
Without repeating the other good answers, I want to add a note about why this line is confusing to an English language learner: It's missing punctuation. Song lyrics are often written without punctuation, or with very sloppy punctuation. With the missing punctuation supplied, it would be
Couldn't keep it in! Heaven knows I tried.
Because song lyrics are written with such sloppy punctuation, don't worry too much about why things are capitalized a certain way. It may very well be wrong. As I wrote in a comment, according to style guides, heaven is not usually capitalized (except for reasons unrelated to the word itself), even when it refers to the Christian heaven.
Heaven knows or God knows is a mild oath, emphasizing the truth of the statement "I tried." It's similar to the stronger oath "(as) God is my witness."
"Heaven knows" and "God knows" are interchangeable.
Capitalization is a bit more tricky. Usually, "Heaven", "Hell", "God", "Lord", etc. refer to the Christian concepts of those and should start with an upper-case letter. However, when talking about general concepts, they usually don't.
Notice the difference in capitalization between:
With God as my witness, I shall exact my vengeance on you.
and
Zeus was a god worshiped in Ancient Greece.
Max's and LarsH's answers are solid, but they're missing a nuance of the difference between "Heaven knows" and "God knows" which should be made clear.
Theoretically these expressions are interchangeable, but there is a slight difference in how they could be taken by particularly religious people. There are some speakers who view use of "God" in an idiomatic expression, or in any context in which one is not reverently speaking of or to the deity, to be (at least mildly) offensive. (For discussion elsewhere, see here and here).
Disney, obviously, would want to make its movies as family-friendly as possible, and so will go for the version that has the least chance of offending people.
Another consideration is that in this context, "heaven" scans better. "Heaven" has two syllables, with the emphasis on the first syllable, while "God" has only one syllable. Even more important, it ends on a "d" which would make the word come to an abrupt stop when singing it. So singing "Heaven knows I tried" has the same pattern of emphasized-syllable followed by non-emphasized syllable. "God knows I tried" would either make the singer stretch the word "God" awkwardly, or give an awkward gap in the singer's performance of the line.
I know this is a little beyond what you asked about, but when you're trying to figure out why someone said something one way instead of another, these are always important considerations: both levels of politeness, and the ways that rhythm and sound affect how we use language. These ideas will become more and more important as you continue your journey through learning English!
It has been noted that Heaven refers to a place in the afterlife, but in this case, Heaven means more than that. The word Heaven can also refer to the Being or beings in Heaven; similar to how, in the press, the phrases "the White House announcement" or "Scotland Yard's policies" or "Pyongyang's military brass" could refer to an official statement by an American President or cabinet, or the rules of London's police department, or the military advisors of North Korea's government.
Thus, the word "Heaven", in this context, most likely refers to G-d himself. Also, consider this quote from the ancient Chinese philosopher Mozi. Here, the Chinese word Tian is translated "Heaven", but you can tell that he was referring to the Deity.
Moreover, I know Heaven loves men dearly not without reason. Heaven ordered the sun, the moon, and the stars to enlighten and guide them. Heaven ordained the four seasons, Spring, Autumn, Winter, and Summer, to regulate them. Heaven sent down snow, frost, rain, and dew to grow the five grains and flax and silk that so the people could use and enjoy them. Heaven established the hills and rivers, ravines and valleys, and arranged many things to minister to man's good or bring him evil. He appointed the dukes and lords to reward the virtuous and punish the wicked, and to gather metal and wood, birds and beasts, and to engage in cultivating the five grains and flax and silk to provide for the people's food and clothing. This has been so from antiquity to the present.
Regarding capitalization, the word "God" is always capitalized in a monotheistic context. Thus, "God", "Allah", "Jehovah", and so on. The words "heaven" and "hell" and "purgatory" can be capitalized, as they are proper nouns, the names of specific places, but people frequently fail to capitalize them. It really doesn't matter. On the other hand, people seldom fail to capitalize the name of the Supreme Being.
Heaven knows is an idiom. The meaning comes from these two words together. 'Heaven' doesn't have much or any independant meaning here.
The phrase 'heaven knows' serves to emphasise the statement "I tried [to keep it in]"
See this definition of Goodness/God/Heaven/Christ knows from the Cambridge English Dictionary:
used to mean "I don't know" or to emphasize a statement. Some people may find this use offensive:
