You can design a mixer operating in linear (triode) region.  If you skip placing the constant current source at the bottom of mixer, it means transistors have no DC bias applied to their gates, leaving only AC signals driving them.  This might be useful technique in low voltage applications since Vds consumes smaller headroom.  The drawback is that the input amplitude has to be way larger than Vth.  MOSFET gets to cut off when Vgs<Vth.  In other words if you give AC signals that crosses the area smaller than Vth it will produce crossover distortion.  I wonder how well you can reduce spurious with this type of mixer.

What about Gilbert active mixer?? Can the switching quad operating in triode region?

I think the question that is being asked very specific to a current mode Gilbert cell mixer (single or double balanced).

No, the switching pair or quad cannot operate properly in the triode when it is being used as a periodic current switch. The reasons are the same as for any cascode amplifier. When the common gate devices (switching devices) enter triode their drain current is no longer a function of only the source current (transconductor input). In triode, the drain currents are a function of both the source current and Vds because the output resistance of the switching pair is now Ron since the channel is no longer pinched off [i.e. the output resistance dropped and it is a function of both Vgs (Vlo) and Vds (Vout)].

Your (conversion) gain drops dramatically because of the drop in output impedance.

I find it helpful to monitor Vds and Vdsat of the switching core and transconductor devices when I working on transmitters.

Hope this helps.

On a side note: I have always accepted the term "passive" mixer for FET voltage or power mode mixers operating in triode. But I have a hard time completely accepting the term "passive" since the channel is inverted even if it is not operating in pinch off (saturation). The gate is still controlling the flow of current, so technically it is still an active device. Just because it does not require DC drain current doesn't mean it is not an active device in an active mode. I guess we would have to start picking nits about the definition of "active" versus "passive."