This achievement is quite a tricky, pain in the arse, but if you're along far enough in your playthrough it shouldn't be to tricky.
There are three ways to get quests:
1. Someone randomly shows up at your door asking for items.
2. You randomly meet or spot a quest giver on the map.
3. You can scan frequencies on the radio; and this will yield one of three results: the location of a trading caravan, a quest giver, or nothing.
Quest givers show up on the map as a yellow circle with an arrow in it; trading caravans also show up in like this, and are also yielded from scanning the frequencies. However, these do not count.
Complete any 10 of these quests, repeats count, and the achievement should pop.
Just a hint about quests is that they often involved acquiring "x" amount of something and giving to the quest giver. So before going for this achievement I would make sure that you've built and upgraded your bunker. I was in about day 150 before I started seeking these out. The other types of quests involved going somewhere and killing someone. I only had two of these.
NOTE: Someone has commented that the trading caravans DO count for them after this last update. I have not played or tested it since the last update.