The short answer is “Yes.” However, there are a number of caveats involved. Regardless of whether you use SketchUp or some other modeling software, you need some sort of 3D inferable data to start with. The photo of the dog, by itself, contains no inherent 3D data … while my mind’s eye can see the bas-relief image of the beagle you want, it’s doing so based on my 3D knowledge of beagle geometry (so-to-speak).
That being said, I found a number of photo -> 3D model converters on-line. However, due to the lack of definitive 3D (height/elevation) data in the photo, this was about the best result I could achieve:
It’s worth noting that this cannot be carved out using SketchUcam … some other 3D CNC milling software would be needed to actually create it.
If this happens to be your own beagle, you might want to look into using a cheap dual lens 3D camera to take a reasonably close-up picture. The software available to use the two images and infer the 3D should produce a sufficiently detailed model to meet your needs. A more expensive approach would be to use a 3D scanner of some sort. You can buy your own or perhaps there’s a local shop that can scan the dog for you.
[added] You could always use the photo and manually shape a mesh into a beagle shape. If this is the route you take, then I don’t believe SketchUp is the best tool for this type of sculpting.