Define a random field $W$ so that:
- $W(t,T)$ is a standard Brownian motion for every $T$.
- $dW(t,T_1) \, dW(t,T_2)=c(t,T_1,T_2)\,dt$
- $W(t,\cdot)$ is a a continuous function for every $t$.
Suppose $f$ is a deterministic function. I want to find the covariance $$ \operatorname{cov}\left(\int_0^t f(s,T_1) \, dW(s,T_1), \int_0^tf(s,T_2) \, dW(s,T_1)\right). $$
When $W$ is a BM instead of a random field, we can use $XY=\frac{1}{4}[(X+Y)^2-(X-Y)^2]$ to find the covariance but I don't think it is possible in this case. However, my guess is that
\begin{align} & \operatorname{cov}\left(\int_0^tf(s,T_1)\,dW(s,T_1), \int_0^tf(s,T_2) \, dW(s,T_2)\right) \\[8pt] = {} & \int_0^t f(s,T_1) f(s,T_2) c(s,T_1,T_2)\,ds. \end{align}
However, I might be wrong but even if I am right I don't know how exactly to prove this.