Sending complex signal over a single wire
Note that signals sent over wires (and over the air, and over any medium) are always real. What quadrature means is that, on a passband channel (wired or otherwise), you can transmit two signals at the same time. The first signal, which we'll call $s_I(t)$, is mixed with a carrier $c_I(t)=\cos(2 \pi f_c t + \phi)$; the second signal, $s_Q(t)$, is mixed with $c_Q(t)=\cos(2 \pi f_c t + \phi + \pi/2)$, where $f_c$ is the center frequency of the channel.

Note that it is important to use a passband channel. Quadrature communication is impossible on the baseband channel. Maybe this is what confused you, since one tends to associate wires with baseband. However, as long as you don't exceed the bandwidth of the wire, it's perfectly possible to transmit modulated signals over them. In fact, in telephony, the very first trunk links used single sideband modulation to multiplex several phone calls over a single wire -- this was before PCM and digital telephony were even invented.

The transmitter would then calculate the following signal: $$s(t) = s_I(t)c_I(t) - s_Q(t)c_Q(t),$$ where the negative sign is just a convention. Assuming the carrier phase $\phi=0$ for simplicity, this signal can also be written as $$s(t)=\text{Re}[(s_I(t)+js_Q(t))e^{j 2 \pi f_c t})].$$ This expression is sometimes useful in a digital implementation, since you only have to keep track of two complex signals instead of four real signals.

The channel adds noise, so the received signal is $r(t) = s(t) + n(t)$. The noise is usually modeled as Gaussian, white noise with PSD equal to $N_0/2$.

In the receiver, we have (ommitting a lot of algebra readily found in textbooks and on other answers in this site): $$\hat{s}_I(t) = s_I(t) + n_I(t) = \text{LPF}[r(t) c_I(t)]$$ and $$\hat{s}_Q(t) = s_Q(t) + n_Q(t) = \text{LPF}[r(t) c_Q(t)],$$ which means that the receiver can recover both transmitted signals (albeit with noise).

How to actually get all this to work in an actual system requires solving other problems that have not been addressed here and that can take a whole textbook to explain. I recommend that you start with this free textbook: http://sethares.engr.wisc.edu/telebreak.html (link at the end of the page).