Usage of "make up my mind"

When you are not sure about a certain usage of a phrase, it is good to use corpora such as Corpus of Contemporary American English (COCA) to find out how the phrase is being used in novels, magazine articles, etc.

In this scenario, it is not wrong to use make up my mind since ODO suggests that it has the definition:

make up one's mind

make a decision; decide: He made up his mind to attend the meeting.

and COCA shows that the phrase make up my mind about something has actually been used though a better way to write it would be

I made up my mind to quit smoking.

The word resolve has the meaning

resolve

decide firmly on a course of action

so if you want express that I am determined to quit smoking then resolve is a better word.

And yes, it would be better to write it this way:

I resolved to quit smoking.

Yes, both seem to be correct in this context. Though "make up my mind" would suggest you had conflicting opinions within yourself and you had to spend quite some time deciding on which. "Resolve" seems to suggest a decision without having spent much time with different sides of the matter.