What performance should we look at for a good transconductor stage design?

If I want to design a good transconductor or Gm-cell for a Gm-C filter, what performance should it have?

If I want to use it to build a 2nd order LPF which cuts off at 40 MHz, what bandwidth should the Gm-cell have?

Also, people usually only talk about transconductance of the Gm-cell, but what voltage gain should the Gm-cell have?

You can't talk about a design of a Gm cell in general. It is all specific to a design.
However, for a filter, you would want to have that Gm cell perform as an integrator, (when loaded capacitively) well beyond the pole frequency of the filter Maybe at least 10x but even 100x. That means -90deg of phase for the frequencies of interest and beyond. You would also want to have high DC gain. Create a model for the Gm cell with finite DC gain, finite BW and use it to simulate your synthesized filter. Then you can see how each limitation affect the filter frequency response - pole shifts and Q degradation/enhancement. Often Gm-cell for GmC filters are some sort of diff pair with degeneration. The degeneration will introduce zeroes which also influence the filter frequency response through the resulting phase response of the Gm cell.
Then you have to design it also taking into account distortion and noise.
Gm cells can work to pretty high frequencies if you sacrifice on some of the above mentioned limitations. If you simulate a Gm cell of certain size loaded by a similar Gm cell that is diode connected you can get an idea of how fast it can ultimately go.

It is a general question and priority of parameters in spec depends on application.

However, typically you might be constrained by following: input referred noise, linearity (IM3/CP1dB) - both in-band/out-of-band, stop band attenuation and other more common stuff like CMRR or PSRR.

Bandwidth of Gm cell should be "high enough"
Again it depends on application, sometime architecture. Basically, you would like to determine filter parameters by Gm and C of used components, so your Gm should be constant for wide range of signal frequency and amplitude. To find an answer for bandwidth question, you should now the behavior of filter architecture if you start to loosing Gm. So, bandwidth of Gm cell used only as integrator might not be crucial, but for cell used in feedback might be critical. Here, behavioral modeling is a good way to understand the spec.