I have a specific person whom I want my character to marry. I have the amulet of mara, and I have spoken to that guy in Riften to open up marriage dialogue (because other characters will mention marriage), but the character (Sylgja) doesn't give me her "favor" quest. I've snooped around online, and it seems like many of these quests are radiant, and sometimes they just aren't available. Is there anyway (on the Xbox) to get the quest? Should I just keep doing other quests and pop back in occasionally to see if it's changed?

Without using the console you are limited in what you cause to happen. Read the wiki article concerning your intended and do any quests that she offers, and any quests the wiki suggests too.

There may be 2 (or 3) things you have to do for the lovely Sylgia:

thanks for the answers, guys. the problem is that I'm not given any quest options. I just have the default dialogue options.

Have you talked to the other miners/cleared the mine yet?
If not, do so first maybe. If so, then I've got nothing.

if i remember correctly

you take her dad the package.

when there you do a quest for her mother and then take the return package from her dad back to her.

then she should have the option of marriage. it is the quest for her mom that alllows her to marry you.