There are two separate issues here: omitting the proof, and calling it an exercise.
It is relatively common to omit some proofs in research papers. In particular, proofs that are long but routine, or that re-use the same method as other proofs in the paper, are often abbreviated or removed. It depends on the intended audience for the paper, to some extent. You will get a sense of this by reading more research papers. But caution is needed: one of the first places to look for errors in a paper is in results where the proof was omitted!
The question whether to call something in your paper an "exercise" is more difficult. I wouldn't be surprised if some people do it -- but some people don't write very well. The audience for a paper includes your peers, and you are not really in a position to assign them homework. There are other ways to indicate that a proof is easy besides calling it an exercise: "the following theorem follows directly from definitions", "the proof of the following theorem is routine", "the proof of the following theorem is similar to the proof of Theorem 1.3", etc. Giving a small hint of how to prove the result can make the paper stronger, and it helps any readers who are not experts in the area.