I don't hope to gain your forgiveness; I just hope you will feel better after my apology.
I'm not begging your forgiveness ...

Do both of the above sound good to you, especially "gain your forgiveness?" Thanks.
Gain forgiveness is OK.

I don't think the meanings are the same. Not to hope and to not be doing something are different.
Hi Angliholic

I've heard people say 'to beg/ask for your forgiveness' and 'to get/have your forgiveness'.


[edit] Deleted 'I don't think people say 'to gain sbd's forgiveness' because apparently people do say 'to gain forgiveness'. My apologies.
 Mister Micawber's reply was promoted to an answer.