What qualities make someone considered intelligent? Is it problem-solving skills or knowledge?

First question

Although most people think of intelligence as a level of IQ, researchers think of it as psychometric g. Bright people have high g and dull people have low g. IQ is used as a convenient means of measuring, scaling, and comparing differences in intelligence. IQ measures g and some non-g factors, which are present whether the researchers want them there or not. These non-g variances can be thought of as noise.

Second question

Now that we have g as the essence of intelligence, we can look at the factors that combine to make it:

source: Haier, R.J., Colom, R. and Hunt, E., 2023. The science of human intelligence. Cambridge University Press.

This is a three stratum model with g at Stratum III, broad abilities at Stratum II, and narrow abilities at Stratum I. The answer to your question lies in the 8 broad abilities shown in Stratum II.

Problem solving is shown above as Gf (fluid intelligence). Knowledge is shown as Gc (crystallized intelligence). At a given level of g, people tend to do approximately the same in measures of each factor. This property of intelligence is known as the positive manifold and is central to the understanding and modeling of intelligence. It tells us that bright people are not the result of a single factor, but of the full range of factors. The same applies do dull people; they typically show low scores on all measures of the underlying factors. Gf and Gc are very strongly correlated. Both also show high heritability. In fact, g is up to 91% heritable.

It is possible to construct numerous similar structures that each do a reasonable job of modeling intelligence. The most common of these is the three stratum model, largely because it can be modeled and measured by an IQ test that is not excessively large and complex. Additional complexity is of some interest to researchers and little interest to people who are actually using IQ tests for various research and clinical reasons.