According to Cambridge Dictionary,
Adorable means: used to describe someone or something that makes you love or like them, usually because they are attractive and often small.
I was wondering if I can say: It was an adorable time with my family.
I think it's a strange phrase. It's understandable, but it sounds weird, because adorable is usually applied to people or animals or behaviors, or at least concrete nouns, but not times.
If we look at Google Ngrams for nouns used with adorable in the last 20 years, we see that the most common are
As you can see, only one, or maybe two, of those (way, thing) is not a concrete noun.
If we compare "adorable time" to "adorable man/thought/table", to use three words that didn't even appear in the first list, we can see that "adorable time" is barely more common than "adorable table"!
Conclusion: it is just not an idiomatic way to use the word.