Whether things like "intelligence", "cognitive ability", and "aptitude" (some of which may be synonyms depending on your view) are innate vs. learned or fixed vs. variable over time are orthogonal to each other. And for each of those pairs, the answer may not be as simple as a binary division or even a gradient (it may decompose into something weirder, being causally determined by multiple factors where some of those
factors are fixed and others aren't).
Moreover, both of those questions are separate from questions that get at what IQ measures (does it measure aptitude, does it measure factual knowledge, does it measure social knowledge or acculturation within a specific context, etc.).
Lots of things are easy to identify as both substantially genetically determined and variable over time and mediated by environmental factors, e.g., height. Lots of things are likewise easy to identify as significantly environmentally determined but also largely stable over time if not altogether fixed (e.g., personality, attachment styles).
It's also at least possible for all of the following to be true at the same time:
- IQ tests correlate with socioeconomic status
- IQ test scores vary over time and can be increased
- some IQ score increases, or some part of a given IQ score increase, reflects a genuine aptitude increase
- IQ tests are somewhat gameable in that training for IQ tests can increase scores so that some of the measured increase does not measure improved cognitive ability
where aptitude means something like fluid problem-solving ability, speed of learning, etc.