Hi. Please tell me what the difference is (if there is one). I think there is a difference. Thank you in advance for your help.

1. They talked about their goal and (1) their heart's desire/ (2) their hearts' desire.
Heart's refers to one heart, and hearts'refers to two hearts. Emotion: wink

Because your sentence started with 'They', there must be more than one heart, so therefore hearts' desire must be correct. Of course, there may be more than one desire (different for each of them) so that would be their hearts' desires.

The heart's desire form would come if there was only one person, and therefore one heart, as in:

If someone you love wishes that they will get their heart's desire ....

Students: We have free audio pronunciation exercises.

our hearts desire

Our hearts' desire is to be a blessing.