AST Boosted, can't find Job Quests to continue.

You should edit the answer into your post instead of deleting it.

For anyone who stumbles on this thread in the future with the same problem, There are two quests at each cap level. The first gives the cap level skill and ends the expansion's job story; after this, you must finish the expansion's MSQ. Then you can do the second quest for that level, which is the beginning of the next expansion's job quests.

The job boost completes all job quests 1-60, but you won't be able to start the SB job quests until you finish Heavensward MSQ.

In summary, Lv50~Lv60 job quests require completion of ARR MSQ (2.55). Lv60~Lv70 job quests require completion of HW MSQ (3.56).