Quick triangle search for AABB-triangle intersection test

I guess not. You will have to test all your triangles separately, except you triangles have something in common which would allow to simplify the intersection test by performing one of its steps for multiple triangles at the same time.

As to your comments: "... is [it] possible to select the nearest triangle to the bounding box and perform the intersection test?". How would you define nearest? If you define it as: distance between the triangles center and the bounding box's center: No, that doesn't help you. You can easily construct two triangles and have the nearer one not intersect and the further one do well intersect. If you defined the distance as: The closest distance of any point of the triangle surface to any point inside the bounding box, then you are basically performing the same test as for your intersection test.

So, in a nutshell, no, there is no shortcut. But you could work on the speed of your intersection test, which you haven't shown here. If you are dealing with millions of triangles, most of which fall wide outside the range of the bounding box, your intersection test should be structured, such that these trivial cases are handled in minimum time. For example, you could place the test that are aligned along the coordinate grid up first, so that non-overlaps appear within the first couple of test steps.