Jerry rig a common wire from another zone?
You just can't get what you want...
Unfortunately, each zone relay has its own independent 24VAC transformer, so trying to &quot;poach&quot; a C wire connection from another zone's relay simply won't work at all.  No sense continuing down that dead end!
However, you just might be able to get what you need
The good news is that there is a way out of your situation, since the Ecobee model you have supports two-transformer systems.  You'll need:

a faceplate mount or knockout mount 24VAC Class 2 transformer (if you're using the latter, you'll need a suitable faceplate for the box)
a few feet of 18/2 thermostat wire
a few feet of 12/2 armored cable with matching bushings (&quot;redheads&quot;) and cable clamps
Crimp spade lugs suitable for 12AWG wire
Wirenuts
Mounting screws
A suitable tool for crimping the spade lugs on
And a 4&quot; square metal junction box

We first start by turning the HVAC system breaker off and mounting our 4&quot; square box somewhere convenient near the existing zone relays, then running the 12/2 armored cable over to said 4&quot; square box from the top of the leftmost zone relay using the &quot;redhead&quot; anti-short bushings and cableclamps.  Once that's done, we can wirenut the transformer's 120V leads to the black and white wires on the armored cable, then mount the transformer.  At the other end of the AC run, we crimp spade lugs onto the white and black wires then land them on the appropriate terminal screws on the zone relay board.  Once those terminal screws are retightened, then we can move onto the secondary side of the transformer.  The 18/2 cable attaches to the two terminal screws on the transformer at one end, and gets wirenutted to the green and blue wires on Zone 1's thermostat cable at the other end.
Now that we're done at the zone relays, we can tackle installing the new thermostat for zone 1.  Green and blue go to Rc and C respectively, while red and white go to Rh and W/W1.  This way, the Ecobee will draw power from the 24VAC transformer you just installed, while controlling the heat by connecting Rh to W to activate the zone relay by bridging the T-T terminals on said relay.