One thing which is really useful for this type of project is the ability to do very quick ballpark calculations to see if something is (a) easy with no special effort, (b) possible but difficult with off-the-shelf parts, or (c) completely impossible without government-lab levels of funding.
In this case, with a transimpedance amplifier, one thing besides the amplifier that limits the bandwidth is going to be the capacitance in parallel with the feedback resistor. Think of an inverting amplifier with a capacitor in parallel with the feedback resistor, as a low-pass filter (see
this at the "Frequency Response Curve" section, or work through the math from an ideal op-amp), then replace the input voltage across input resistor (which acts as a current source) with a pure current source, and you have a transimpedance amplifier. You'll find that the feedback resistor and its parallel capacitance adds a pole to the frequency response at 1/(2*pi*Rfbk*C).
Anyways, that was just showing you where my math is coming from. Let's assume the op-amp's bandwidth isn't even a problem, and look for what capacitance is allowed in parallel with the 1Gohm resistor to give you a pole at 300 Mhz or higher. Knowing that 300 Mhz=1/(2*pi*1Gohm*C), the parallel capacitance has to be < 5.3E-19 F, or 0.00000053 pF! Considering that capacitances like 0.01pF - 0.1pF are seen just by traces on the same board, expecting to achieve 0.00000053 pF of stray capacitance across your feedback resistor is wildly unrealistic. Even if you made the resistor a mile long for almost zero stray capacitance, there's way more capacitance than that just between the op-amp's pins.
So maybe it would be more realistic to have multiple cascaded stages, with lower gain in each one? In that case, each one would need > 300 Mhz bandwidth, but just as an easy example let's suppose we can survive with 300 Mhz bandwidth in two separate stages. Let's try and split up the gain between the stages, with the first stage having a gain of 1E4 and the second stage having a gain of 1E5:
2-stage exampleNow with a 10K feedback resistor in the first stage (a transimpedance amplifier) you'll need "only" < 0.05 pF across that resistor. This is still difficult, and will require some special feedback-resistor-strings like DaJMasta mentioned, but at least it isn't wildly unrealistic.
The second stage with a gain of 1E5 will be a voltage amplifier (because your transimpedance amplifier is now producing a voltage output), so let's suppose that an inverting amplifier: if you use an input resistor of 1 ohm (you can do that here because the input voltage will be so tiny that the current through 1 ohm will still be very small), your feedback resistor will be 100K. To give the second stage a bandwidth of 300 Mhz, it can't have more than 0.005 pF (5 fF) across its feedback resistor...uh oh, this is starting to look unrealistic again, especially because to get 300 Mhz overall bandwidth, you'll need > 300 Mhz bandwidth in each stage.
3-stage exampleMaybe splitting it into 3 separate stages will help, each with a gain of 1E3. I did a quick LTSpice simulation, and it turns out that 3 stages of 600 Mhz bandwidth each gives you a total bandwidth of 300 Mhz (see attached image).
So the first stage will have a 1K feedback resistor, and its stray capacitance will have to be 1/(2*pi*600E6*1E3) = 0.27 pF or less. The second and third stages can then be voltage amplifiers with input resistors of 1 ohm, and feedback resistors of 1K, with the same stray capacitance requirement on each one (< 0.27 pF). This is sounding fairly realistic.
Even though using a 3-stage solution this way solves the stray-feedback-capacitance problem, it adds a noise issue though. A 1-5 pA input will give you 1-5 nV at the input to your second stage. This is an absolutely tiny signal, and even the lowest-noise op-amps have input-referred noise levels of about 1 nV/sqrt(Hz). I don't want to overwhelm you here by going into a full noise analysis tutorial, but basically if you solve your stray-capacitance problem, your signal is going to be completely swamped in noise (even with 2 stages probably). There are measurement instruments that can measure nV-level signals, but that is usually done with methods that
don't allow high frequency (chopper amplifiers, heavy averaging, etc.).
Anyways, I hope that wasn't too much information

Just wanted to:
1. Show how someone else would think about a problem like this, and
2. Demonstrate how useful some quick feasibility calculations are - it's much easier to spend an extra day doing some easy math, than to order parts and build things immediately, before finding out that what you're trying is impossible by a factor of a million!
My best advice overall is that you won't be able to have both low current (single pA) and high bandwidth at the same time. See if there's a different way to make the application with only one of those things.