I guess what you are looking for is a product rule (used for partial integration). For such problems, it is usually very helpful to write the expression explicitly (in coordinates). We have $$\vec a \cdot \Delta \vec b= \vec a \cdot (\vec\nabla \cdot \vec\nabla) \vec b = \sum_{i,j}a_i \partial^2_j b_i.$$
Now take a look at $$ \sum_{i,j}\partial_j (a_i \partial_j b_i)= \sum_{i,j} \left( a_i \partial^2_j b_i +(\partial_j a_i) (\partial_j b_i) \right).$$
The formula for partial integration thus reads $$\int \!d^dx\,\vec a \cdot \Delta \vec b = \underbrace{\int\!d^dx\,\nabla\left(\sum_i a_i \nabla b_i\right)}_\text{surface term using Gauss} -\int\!d^dx \sum_i(\vec\nabla a_i)(\vec\nabla b_i). $$
Some more (potentially) useful formulas:
Interchanging $a$ and $b$, we have $$\int \!d^dx\,\vec b \cdot \Delta \vec a = \underbrace{\int\!d^dx\,\nabla\left(\sum_i b_i \nabla a_i\right)}_\text{surface term using Gauss} -\int\!d^dx \sum_i(\vec\nabla a_i)(\vec\nabla b_i). $$
Subtracting the two relations yields (this is the vector version of Green's second identity) $$\int \!d^dx\,(\vec a \cdot \Delta \vec b- \vec b \cdot \Delta \vec a) =\int\!d^dx\,\nabla\left(\sum_i a_i \nabla b_i - \sum_i b_i \nabla a_i\right).$$
Adding the two relations yields $$\int \!d^dx\,\vec a \cdot \Delta \vec b = \int\!d^dx\,\nabla\left(\sum_i a_i \nabla b_i + \sum_i b_i \nabla a_i\right)- \int \!d^dx\,\vec b \cdot \Delta \vec a-2\int\!d^dx \sum_i(\vec\nabla a_i)(\vec\nabla b_i). $$