Like any simple-seeming cognitive sciences question, it is important to start with a series of disclaimers. It might seem like human intelligence or intelligence more generally is an intuitive concept, but once you start to explore your intuition or look at historic definitions of intelligence, you see that intelligence is a very ill-defined and slippery concept. You could, of course, take the behaviorist view, and define intelligence as performance on IQ-tests, and then your question has a simple answer: no, we can't measure that in most animals.
Alternatively, you might try to look at general behaviors that you usually associate with intelligence, then you can often come up with tests for animals. However, this might challenge some of your preconceptions. I want to provide two example tests (but many many more exist) to illustrate my point.
When we usually think of 'intelligence' we are eager to associate things like behavioral flexibility and innovation with it. Both of these can be tested.
Serial discrimination reversal learning
Serial discrimination reversal learning is a relatively standard procedure for studying animal cognition. The first reference I can find to this specific methodology is from 1966 in the sea lion (Schusterman, 1966), but the basic ideas of using similar tests in animals was already common practice before WW2 (Spence, 1936).
The task is usually given in the form of a choice between two actions (say pushing a button). At first whenever the animal clicks the left button, it is given a treat, and nothing for the right button. This training phase is continued until the animal learns that left button gives food. At this point, the reversal phase is enacted, and suddenly the left button stops giving food, and the right button does, and the animal is trained to associate the right button with food.
These learning, reversal, learning cycles are repeated several times and the rate of change from one button to the other after reversal, and consistency of picking the correct button during learning are tracked to give a measure of the animals behavioral flexibility. There is great inter-species variation on this, but also a lot of intra-species variation.
Innovation tasks are popular in the context of social learning, where the animal (and the species) has to balance between innovating and copying behavior (Reader & Laland, 2003). The classic example of this is English greats tits learning to break the foil caps of home-delivered milk bottles to eat the cream at the top (Hawkins, 1950).
An example I want to highlight is from Griffin et al. (2013), where myna birds where tested with a novel extractive foraging task. In this task, birds faced difficult foraging tasks like having to pull out a piece of paper from a champagne flute before getting food, or by having to open a dish with a lid that could only be lifted by a hook in the middle and not from the side. Birds show great interspecies and intraspecies diversity on these sort of innovation tasks.
Naively, one might expect both reversal learning and innovation to be important components of intelligence, and 'smarter' animals to do better on both. This intuition would be well supported if you looked at the inter-species level, you would find an increase in innovation and behavioral flexibility with relative brain size. However, at the intra-species level, Griffin et al. (2013) show that this is not necessarily the case. Among myna birds, those that do better on the innovation task, tend to perform worse on the reversal learning, and those that do worse on the innovation task, tend to perform better on the reversal learning. This means that if you are looking for variation within a species, then you have to be careful with your tests. Two tests that both correspond to an intuitive notion of "intelligence" in humans, might in fact correspond to opposite ends of the spectrum in the non-human animal.
Griffin, A. S., Guez, D., Lermite, F., & Patience, M. (2013). Tracking Changing Environments: Innovators Are Fast, but Not Flexible Learners. PloS One, 8(12), e84907.
Hawkins, T. (1950). Opening of Milk Bottles By Birds. Nature 165(4194): 435–436.
Lefebvre, L. (2011). Taxonomic counts of cognition in the wild. Biol. Lett. 7: 631–633.
Reader, S.M. & Laland, K.N. (2003). Animal Innovation. Oxford: Oxford University Press.
Schusterman, R.J. (1966) Serial discrimination-reversal learning with and without errors by the California sea lion. J. Exp. Anal. Behav. 9: 593–600.
Spence, K. W. (1936). The nature of discrimination learning in animals. Psych. Rev,, 43(5): 427.