How do I unlock 60g or 100g quests?

You've just been unlucky. There is no special action that unlocks these quests or makes them more frequent- they're available from the start.

A 1/500 chance isn't that unlucky. When you consider the thousands of people who play this game, this is going to happen to someone.

EDIT: This answer is limited by the evidence in the Reddit post the OP originally linked to. More recent evidence suggests there are unlock requirements. Read on for a cautionary tale as to why you need larger sample sizes before jumping to conclusions though. :)

I agree with Studoku. However, two things are worth noting:

First, the exact chance of that occuring is just greater than 0.2% as the odds of not getting such quests 30 times is 0.813^30 = 0.002 (roughly; it is slightly higher.)

More importantly, the analysis posted had a sample size that isn't terribly large. Constructing a confidence interval for proportions in the range of [p - 1.96*sqrt(p*(1-p)/n), p + 1.96*sqrt(p*(1-p)/n)] gives us a 95% confidence interval from 11.38% to 26.02% . This means your odds of failure can only be assumed to be (1-0.1138)^30 = 2.67% with 95% confidence.

While this is lower than 5% (for the purposes of frequentist statistics) a Bayesian might suggest that a 2.67% chance of this happening is still more likely than both the odds of only your computer having a bug or Blizzard having secret unlock requirements that no one has figured out yet combined.

Either brand of statistician would probably suggest that the original sample size was far too small, and that more testing was needed.

Explanation:

The formula used to come up with the confidence interval is listed here. A confidence interval is a way of saying, I am X% certain that the true proportion (or mean) is between Y and Z. In this case (as is done commonly), I chose 95% for the confidence. This means, given our evidence, it is 95% likely that the true proportion lies between 11.38% and 26.02% according to that formula.

The probability of NOT getting an event is: 1 - p where p is the probability of the event occurring. The probability of NOT getting an event x times is (1 - p)^x. In this case, I chose the conservative side of the confidence interval as p, 11.38%.

Thus, I used (1 - 0.1138)^30 as the complement of this event happened 30 times. This equaled about 0.0267 which translates to 2.67% because probabilities are defined as being between 0 and 1.

Frequentist statistics are the classical (older) school of statistics that set a particular value (often called alpha) before an experiment such that if the probability of an event (or series of events) happening is lower than the chosen alpha, they will reject an old belief in favor of a new one. In this case, the old belief could be that the original poster was simply unlucky, and the new belief could be that there is some other mechanism at work which is influencing his/her quests. A common value for alpha is 0.05, or 5% probability.

Bayesian statistics is relatively newer and doesn't rely on strict alpha values for acceptance or rejection criteria. Instead, they consider a series of prior distributions, which basically means they consider more of the context of a situation to determine their judgments at the end of an experiment. Bayesian statistics is a bit trickier because calculating the prior distributions can often be difficult or time consuming. In this case, I didn't state the probability of just the poster's computer having a bug but I (lazily) assumed it was extremely low. For a humorous glimpse into these two fields of statistics, see XKCD.