There is, in fact, an extraordinarily useful scientific metric that does operationalize the concept of "intelligence" - most people have heard of it (IQ) but very few people are actually aware of what it is supposed to measure (spearman's g).
Give a sample of people a battery (i.e. a series) of cognitive tests - and this can be virtually any measurable test involving cognitive skills - such as digit recall, word recall, reaction time, stroop test etc, or the more traditional tests of verbal or mathematical reasoning; the more tests you give them the better.
What you will invariably find - and the effect will be more and more pronounced the greater the number of people you give the battery to - is that people who do well on some of these tests also tend to perform well on the others and vice versa (people who are bad at a few of them will tend to be bad at most of them). In other words there is a correlation in your performance in all sorts of cognitive tasks.
There is a statistical technique (factor analysis) that attempts to quantify this universal correlation between people's performance on series of cognitive tasks - the result of that technique is well known in psychometrics and is typically called "g".
In fact, the guy who originally came up with the idea, Charles Spearman, motivated the concept by the observation that school pupils grades on many different subjects were correlated (i.e. school children who get good grades in math also tend to get good grades in not only science but also history, languages etc etc - on average - and vice versa). Spearman suggested that there was some underlying general factor or ability responsible for that correlation (i.e. a general brain processing power or whatever you want to call it).
This essentially is what IQ tests (at least good ones like e.g. the Raven's Matrices) are designed to measure. In fact you can even quantify just how good an IQ test is by establishing "g" with a battery of cognitive tasks and then seeing the correlation between that result and the IQ test - IQ tests that are more correlated with "g" are said to have better "g loading".
Now finally to answer your original question: that school subjects (and hence GPA) are "g loaded", i.e. correlate with "g", is one of the most well established facts of psychometrics. However unsurprisingly not all subjects are equally "g loaded" - some subjects like math and physics, for example, are highly g loaded - others (e.g. chemistry and biology) are slightly less so, and many other subjects significantly less so.
Of course academic performance in the real world is likely a function of both "g" and several other things, an important one which you noted is motivation. This, however, does not mean that general intelligence does not exist and that general intelligence is correlated with academic performance.
