BAQ has been replaced with BAH.. but the following gives a good explaination of what can be paid and why...
Basic Allowance for Housing (BAH) With-Dependents Effective January 1, 1998, housing allowances paid to members of the uniformed services underwent major changes. BAH, which is a combination of the old with-dependent Basic Allowance for Quarters and Variable Housing Allowance will be paid to member s whose dependents reside off-base within the United States, including Alaska and Hawaii. BAH rates are based upon grade, location, and dependency status. The requirement to provide military pay with proof of housing expenses and sharer status has been discontinued. As always, your continued entitlement to BAH depends on the amount of support you provide to your dependents. Questions concerning the amount of adequate support may be addressed to your local financial services office. Finally, if any of your dependents reside in family-type government quarters, you will not be entitled to BAH.
Basic Allowance for Housing (BAH) Without Dependents Effective January 1, 1998, E-4 over 4 years who received partial BAH (partial BAQ) prior to reassignment will receive BAH at the rate for their previous base for the duration of the PCS period. As before, payment of without dependent BAH at Kunsan is contingent upon your approval to reside off-base. If approved, your BAH entitlement will equal the amount of without dependent rate BAQ.
Basic Allowance for Housing (BAH) Diff For single military members assigned to on-base bachelor (dormitory) or off-base quarters with an obligation to pay child support, you should already be receiving BAH-Diff, previously known as BAQ-Diff . With the exception of a drastic reduction to your support amount, you will continue to receive this allowance regardless of where you reside. The amount payable is the difference between the with-dependent and without-dependent BAQ portion of BAH. The requirement to pay child support must be documented on a court order, or a statement signed by both you and the custodian of the child.