you can't arbitrarily have conflicting requirements, and expect valid solutions.
If the voltage source is equivalent a 1 pF capacitor, you will not only have to have an opamp to perform the V/I conversion, but the opamp itself must be a precision instrumentation amplifier with ultra low Ib. Additionally, to prevent surface leakage, the input must surrounded by a guardband driven by a second opamp.
1pF is the capacitance between two poles on both sides of a road, almost nutthin'. There will be no current and you will not be able to measure its voltage. Why bother thinking about it?
A FET (JFET or MOSFET) in common source configuration with internal feedback can act as a voltage to current converter without needing additional quiescent current. Of course it's less linear than an OP circuit.