Yes at DM's discretion
It is not the player's role to suggest what check they want to make. A player should tell the DM what they are trying to do and the DM would adjudicate their request and decide if it is possible and what game mechanic or mechanics (or improvised rules) are needed to adjudicate the results of the attempted action.
Remember that "checks" are a game mechanic that your character is unaware of (even though the you as a player does know about them). It is not an in-fiction thing at all. So, your monk wanting to "make a Dex(athletics) check" means nothing in-game. I think it is always more fun and better for the game overall if players describe what they want to do in-game and let the DM take care of the rules.
The PHB even says as much:
In such cases, the DM might ask for a check using an unusual combination of ability and skill, or you might ask your DM if you can apply a proficiency to a different check.
Note how the PHB says the DM asks, not the player. This is how it should be.
In your example above it might go down like this:
Player: My monk want to try to shove this guy backwards, but since he knows that this enemy is too heavy for him to use brute force he wants to try to push right on his center of gravity to make the shove easier.
DM: OK make a DEX(athletics) check at disadvantage since you aren't quite sure where the center of gravity is.