I know in the English the words of aunt and uncle are used to address our parent's brother and sister. But, I want to know whether there exists a specific lexical item for our (mother/father)'s brother and also for (mother/father)'s sister. For example, when one says my uncle has two sons, I promptly ask whether he is your mother's brother or father's. There is a way to distinguish between them?
There are many languages that differentiate relatives much more than English does. So you can have different words for aunts, uncles, grandmothers, and grandfathers on your mother's side and your father's side. I suspect your native language is one of these.
English doesn't do this. The distinction isn't important enough to require separate words for each. We usually don't bother specifying the relationship any further, and when we do we add the words "maternal" (formal) for our mothers' side and "paternal" (formal) for our fathers' side before the word for the relative. Informally, we just say "uncle/aunt on my mum's/dad's side". However, this doesn't completely define the relationship for aunts and uncles. If I say "paternal aunt" it's either my father's sister, or my father's brother's wife (my father's sister-in-law). (Note that my father's sister-in-law can also mean his wife's sister, who is not related to me if my father's wife is not my mother.)
Some languages also have terms for older brother, younger brother, older sister, younger sister, etc. Using these words shows respect by acknowledging a hierarchy within a family. Not only does English not have the equivalent words, we also have a word "sibling" which is either a brother or a sister without specifying the gender. Many languages don't have a word for "sibling", e.g. instead of asking "Do you have any siblings?" you have to ask "Do you have any brothers or sisters?"
So, in English we have different words depending on the gender of the relative, but not different words for exactly how they're related. Many of our terms are ambiguous without going into more detail. We also have non-gender-specific words for most relations: parent, child, sibling, grandparent, grandchild, cousin. We don't have non-gender-specific words for aunt, uncle, niece, or nephew.
The many comments this answer has attracted shows how the words mean just what they mean, no more, no less. An aunt can be on your mother's side or your father's side. She could be related to you by blood (one of your parents' siblings) or she could be related to you by marriage (one of your uncles' wives). English doesn't specify any further than that, and usually it's not important enough to clarify the situation.
Your language might not have words for "one parent's brother" or "one parent's sister". A native English person learning your language might find this strange but they would cope after a while. Not having more specific words than "aunt" or "uncle" might seem strange to you, but it's the norm in English to not be more specific. Our language is influenced by our culture and our culture is influenced by our language. Apparently, there is no cultural or linguistic need in English for more specific words.
Both of my parents are only children, so I have no natural aunts or uncles. Everyone I call "Aunt" or "Uncle" is actually a great-aunt or great-uncle (my grandparents' siblings), or a cousin of an appropriate age (i.e. roughly my parents' age).
Something about different languages is that although they can all express the same ideas, often there are no direct translations. This is one of those cases. The most natural way is to just say "aunt" or "uncle". Only specify it further if it's important to do so.
One more phrase you can use, "on my (mom/mother)'s side" or "on my (dad/father)'s side". This is less formal than maternal/paternal, and understood by more people.
So you can say, "Uncle John, my uncle on my mom's side, passed away over the weekend, so that side of the family is all flying in for the funeral."
Other posters have covered the common, Germanic way to express this idea ("on my X's side") and the more formal, educated, Latinate way ("my paternal" or "material Y"). There are some other still rarer forms you can use. Basically, you can use (and people have historically used) almost any Latin or Greek word for family members in an anglicized form, if you don't mind people not understanding you and needing an explanation.
An agnate or agnatic relative is one tracable only through the male line; they're a little more specific than paternal since (e.g.) they wouldn't cover your father's sister's husbands. Such relationships are patrilineal, but you wouldn't normally use that word to describe the person himself. This mostly only shows up in old legal English.
Similarly, an enate or enatic relative is one tracable only through the female (matrilineal) line and wouldn't describe your mother's brother's wife. This almost never shows up at all; old legal English used cognate to describe any relation on one's mother's side. In present-day English, cognate is mostly only used to describe historically related words, rather than people.
You can distinguish blood relations from relations by marriage, but you don't normally call someone your blood aunt. Similarly, it's very uncommon to say aunt-in-law. You could say someone is my aunt by blood, but it'd be more normal to just call her my aunt and distinguish other women as my aunts by marriage. Similarly, she's an in-law is mostly used only for the relatives of your spouse and not other people related by marriage elsewhere in your family tree.
In other words, like @CJDennis's answer said, there are ways to express every familial relationship in English, but we mostly don't bother to do so. There's no important hierarchy of status like in Chinese, so there aren't dedicated words for each one. You have to just explain the relationship.
Side point: Many ESL speakers will explain the relationship as a chain: He's my dad's brother's husband. That's clear and gets your point across, but it's also a little unnatural for native speakers because it implies there's no relationship to the speaker. She's my new mom versus She's my dad's new wife imply very different relationships: the second implies that the speaker (at minimum) doesn't feel very close or loving towards the woman. The way to express your relationship to family members you accept as family is not to use a chain but to explain who they are from your own perspective: he's Uncle Bob's husband or he's the partner of one of my uncles on my dad's side of the family.
That's why we say it that way, if you were curious.