Incompressible fluids are of current interest. Considering a (3+1)-dimensional variable-coefficient Boiti-Leon-Manna-Pempinelli model for an incompressible fluid, we perform symbolic computation to work out a variable-coefficient-dependent auto-Backlund transformation, along with two variable-coefficient-dependent classes of the shock-wave-type solutions. Our auto-Backlund transformation is different from the recently reported bilinear one.