If my boyfriend left his dog in my care for the last 4 months and has never returned for him, can I legally keep the dog?

No, unless you write him and ask him to pick up the dog or give him to you. Tell him that you will have the dog put to sleep if you don't get an answer in 30 days. He may or not believe that last statement, but hopefully it will make him make a decision. If he agrees to give you the dog, confirm it in writing (email would be sufficient).