They are pronounced the same. In spoken french, you need context to tell them apart.
It's a very good question and one that opens a new world of grammar for many people. There are basically two kinds of verbs: action verbs and stative verbs. "Action verbs describe actions we take (things we do) or things that happen. Stative verbs refer to the way things 'are' - their appearance, state of being, smell, etc. The most important difference between stative and action verbs is that action verbs can be used in continuous tenses and stative verbs cannot be used in continuous tenses." http://esl.about.com/od/grammarstructures/a/g_stative.htm
This exercise is about the act of doubting, rather than possessing doubt, so your sentence is not the most direct translation.
Larousse indicates that "to doubt" can mean "douter" and "avoir des doutes", depending on the context. Your sentence uses "avoir des doutes" instead of the verb "douter"
English and French use the noun "doubt(s)"/"doutes":
- I have my doubts about him = j'ai des doutes sur lui; j'ai des doutes à son sujet
- her honesty is in doubt = on a des doutes sur son honnêteté
Both English and French use a verb:
- I doubt it = j'en doute
English version uses the noun "doubt(s)" while French version uses the verb "douter":
- she has her doubts (about) whether it's true = elle doute que cela soit vrai
- I have no doubt [or: doubts] about it = je n'en doute pas