How do you know if you are competent in your job?

I don't know about Tibetan Buddhism unfortunately.

To me, "Buddhism teaches humility, humbleness" means the following:

• Beware (be aware of and avoid) the kind problems and behaviours described here: Wikipedia: Māna

  Māna (Sanskrit, Pali; Tibetan: nga rgyal) is a Buddhist term that may be translated as "pride", "arrogance", or "conceit". It is defined as an inflated mind that makes whatever is suitable, such as wealth or learning, to be the foundation of pride. It creates the basis for disrespecting others and for the occurrence of suffering.

  Apparently this is in the Theravada and Mahayana doctrines.

• Conversely "beginner's mind" is seen as a virtue -- Wikipedia: Shoshin in Zen Buddhism

• I've considered posting a question on this site to ask about "self-confidence" -- because that seems to me kind of necessary or helpful for working -- but I decided not to:

  • Theravada talks about "confidence in the Buddha, Dhamma, Sangha", and maybe that's "verified confidence" -- and perhaps that's enough, and is the right kind of confidence -- and what's called in the West "self"-confidence is not quite right
  • Theravada has related terms like Wikipedia: Vīrya -- which may be akin to self-confidence and are beneficial but which don't require any unwise reliance on "self"

• This answer related to Tantrayāna (Tibetan Buddhism) perhaps suggests:

  • Something like confidence or self-confidence is indeed beneficial
  • It's something you can practice

Also as suggested in a comment, "ask your boss" is a partial answer (but not exclusively a Buddhist answer) -- whether you're (seen as) "good at your job" isn't entirely a matter of "self-evaluation" -- and might also be judged (subjectively) by other people who know your work, or judged (more objectively) by various more-or-less imperfect "performance metrics" defined by managers.

You might also set metrics for yourself -- "how well did I accomplish what I tried to do?" -- to the extent that you're a "manager" of your own behaviour.

A more-explicitly Buddhist metric might be something like, "how much gap is there between what's wanted and what is/was" -- because the size of that gap might be a measure of "craving" and "suffering" and is what you're trying to minimise.

From a more Mahayana perspective a related question might (I'm not sure) be something like, "how skilfully, how effectively, did I behave with other people?"

Doing a good job starts with proper attention and words of another:

AN2.126:1.3: The words of another and proper attention.
AN2.126:1.4: These are the two conditions for the arising of right view.”

Doing a good job, we ask questions:

AN3.20:6.2: It’s when from time to time a mendicant goes up to those mendicants who are very learned—knowledgeable in the scriptures, who have memorized the teachings, the monastic law, and the outlines—and asks them questions:

Doing a good job, we work in harmony with others by practicing self-effacement:

MN8:12.1: Now, Cunda, you should work on self-effacement in each of the following ways.
MN8:12.2: ‘Others will be cruel, but here we will not be cruel.’
MN8:12.3: ‘Others will kill living creatures, but here we will not kill living creatures.’
MN8:12.4: ‘Others will steal, but here we will not steal.’
MN8:12.5: ‘Others will be unchaste, but here we will not be unchaste.’
MN8:12.6: ‘Others will lie, but here we will not lie.’
(...omitted for brevity)

Doing a good job continuously requires proper attention:

DN34:1.2.17: What one thing leads to distinction?
DN34:1.2.18: Proper attention.

Doing the best job requires working to the benefit of all. Instead of asking "am I doing a good job?", we might ask "what work can I skillfully do now to benefit all?"

AN4.95:4.2: In the same way, the person who practices to benefit both themselves and others is the foremost, best, chief, highest, and finest of the four.