On average, you earn 12,300 XP per week if you do all your quests: 6000 XP from weekly quests and 6300 from daily quests (7 daily quest with an average reward of 900 XP).
You need 196,200 XP to hit level 50.
196,200 / 12,300 = 15.95 weeks, round up to 16 weeks.
16 weeks is four months, which is the length of an expansion in Hearthstone.
The progression track resets at the beginning of each expansion.
Ergo, you can earn enough XP to hit level 50 from quests alone. Sure, you have to do all of them, but you can.