My professor presented a problem that I solved. He says I can send it to a conference. What is the etiquette in cases like this?
There is no ethical issue. You should be proud of your accomplishment and publish the paper. It may be, in fact, that the professor saw the problem but not the solution and needed it himself. 

In fact, it is fairly common for a dissertation advisor for the doctorate to provide the problems to his/her students that result in their degrees. That isn't universal, but happens regularly in some fields. 

On the other hand, in some fields (mathematics, say) the ability to ask the right questions is extremely important - more so than finding the eventual answers in many cases. But to be a complete researcher you need to be able to do both. However, it isn't expected that you get this ability naturally. So, in a way, the professor is "kick starting" you on your way to being a good researcher. 

Of course, you can also say in the paper that the problem was proposed by Professor X of University Y. The professor can give you advice about the advisability of that, of course. 

Moreover, the question posed by the professor got you thinking. You got to thinking about the solution. But I also guess that it got you thinking about the next question that needs to be asked. If so, you are on your way. 



I should also note, for completeness, that there have been occasions in which someone has posted important unsolved problems as "exercises" in books and elsewhere. This has sometimes been done without comment. The idea is to get the problem out to a wide variety of people who know the topic and might, somehow, have the insight to solve it. Don Knuth, for example, has done this in Computer Science:The Art of Computer Programming.