Both are correct, but number two needs more information, it doesn't work on its own. Number one is fine as it is. To use the second you need to specify when you will give up smoking. For example 'I promised my father that I will give up smoking *by next week/by christmas/after I finish this one!*' but you can't use it on its own or it sounds a little strange! If I was going to say it in conversation I'd say 'I promised my dad I'd give up smoking(/cigarettes).'

Use "would". Stay consistent in the tenses.

second one is correct

Alex Cracknell is correct! "Will" is more certain than "would". Are promises certain? If you say "I promised I would give up smoking", no. But if you have a plan, eg. "I promised I will give up smoking this year", it sounds much more certain! Remember: context is everything!

I would like to add to what Alex said that even if the date on which the speaker in the example has to accomplish his/her promise comes AFTER the moment he/she is speaking, I think he/she (the speaker) could still say 'WOULD' instead of 'WILL', provided he/she has ALREADY DECIDED, at the moment of speaking, that he/she will NOT accomplish his/her promise. That would make unimportant the fact that the date is still to come (at the moment of speaking). That's the way we would speak the sentence in Italian, in the case I mentioned (using our conditional tense, of course), however I will wait for an answer from a native English speaker to know if in English it would be the same as in Italian.