TPS63070: IQ at PWM mode

IQ does not change when you force the device in PWM, since IQ does not include the current required to drive the power switches and the switching losses. You can find more details about the IQ spec in this application note: http://www.ti.com/lit/pdf/slyt412

Since in forced PWM the device is continuously switching the application input current at no load significantly increases. You can get an indication of the no load input current by checking efficiency data. At this device the no load input current changes with the input and the output voltage.

I would not expect a change in total quiescent current flowing into VIN and VOUT pins going from 6V input in buck mode to 3.8V input in boost mode since the control circuit is still the same and supplied with a bias regulator, but I would expect that part of the current flowing into VIN pin will move to the VOUT pin since both are supplying the internal bias regulator and the voltage at VOUT will become the higher voltage.
Regarding the no load application input current, the current into VOUT will appear higher since it needs to go through the power stage and must be converted to a higher voltage. Boost ratio and power conversion efficiency make the quiescent current into VOUT appear higher at the application input.