Situation: if someone on the forum decides to change his/her username, will quotes which are made for their old username,  change as well? 

No.

How to test:

1. Create a new topic as User A
2. Reply, with a quote, as User A
3. Change display name for User A to User B
4. Check.

1 hour ago, Martin1980 said:

Situation: if someone on the forum decides to change his/her username, will quotes which are made for their old username,  change as well? 

Unfortunately, no, because quotes and mentions uses Display Name instead of ID's. 

On 9/20/2015, 7:52:49, Mr 13 said:

Unfortunately, no, because quotes and mentions uses Display Name instead of ID's. 

Hmm. Would seem to be a nice feature to have it target member id

It changes in the author pane of the post but not in quotes which doesn't make sense. It makes sense to make it retroactive in both or none. Change username, show old username in old posts and quotes or new username in old posts and quotes. So at least the posts match up with the respective quotes, which they don't.

A flaw in pretty much every forum software.