Think of almost as meaning roughly 90%. You could replace it with 90% in these examples:
The performance was almost perfect
The cat was almost dead when we found it
He answered almost truthfully
Note that these adjectives and adverbs are all non-gradable. It is possible to use almost with gradable adjectives, but it is unusual.
According to the Cambridge Dictionary, quite has two meanings. If you use it with a non-gradable adjective or adverb, it means 100%, and if you use it with a gradable adjective or adverb, it means roughly 60%. You could replace it with 60% in these examples:
The performance was quite good
The cat was quite hungry when we got home
He answered quite sensibly
competent is a gradable adjective (you can say that somebody is very competent), so quite in this context means 60%.
If you say almost quite competent, it's like saying 90% 60% competent... it's not a sensible way of describing something. In addition, since 90% of 60% is 55%, you could just use quite on its own to mean 55%, as the percentages I have suggested are only approximate.
If you use almost quite with a non-gradable adjective like dead, it's like saying 90% 100% dead: again, it's not a sensible way of describing something.