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The function coefficients may help you, though it is not exactly what you requested for at least three reasons:

sage: e = trm.expand()
sage: e.coefficients(sqrt(2))
[[2*sqrt(2)*c + 3*a + c + 4*d, 0], [3*b + d, 1]]

The three problems:

1. It is not written in the same way as you want, but you can obtain a closer expression like this:

   sage: sum([cc[0] * sqrt(2)^cc[1] for cc in c])
   sqrt(2)*(3*b + d) + 2*sqrt(2)*c + 3*a + c + 4*d
   

2. The second, more disturbing problem, is that coefficients seem not to produce what we would like in this case. In particular, we would like to have

   sage: e.coefficients(sqrt(2))
   [[3*a + c + 4*d, 0], [2*c + 3*b + d, 1]]
   

   I do not now why this is not the case. It may be a bug.

3. Finally, this works for only one square root as such. Maybe some modification can yield the result with several square roots.