With the rapidly increasing proton-proton collision data set provided by the LHC, the ATLAS experiment gains access to Higgs bosons produced with ever higher transverse momenta $p_{\text{T}}^{H}$. Measurements in this phase space are well motivated by a vast variety of BSM models which predict effects that scale with the square of the involved energy scale. The associated production of a Higgs boson $H$ with a heavy vector boson $V$ contributes significantly to the total production cross-section at high $p_{\text{T}}^H$. Combining this production mode with the most prominent decay into a pair of b-quarks promises a large enough signal yield in this rare topology. A novel measurement of the production cross-section times the decay branching-fraction for this process is presented, based on data collected between 2015 and 2018 at a center-of-mass energy of 13 TeV. Results are provided in two fiducial regions of the transverse momentum of the vector boson: $250~$GeV$< p_{\text{T}}^{V}<400~$GeV and $p_{\text{T}}^{V} \geq 400~$GeV. These cross-sections are furthermore interpreted as limits on the Wilson coefficients of a Standard Model Effective Field Theory.