We consider a financial market with three assets: a zero-coupon bond of maturity $T_1$, a second one with maturity $T_2$ and the money market account $B_t$. Assuming the market's risk-free rate $r_t$ is normally-distributed, the spot dynamics of the assets under the risk-neutral measure $Q$ are given by:
$$\begin{align}
\frac{\text{d}B_t}{B_t}&=r_t\text{d}t
\\[6pt]
\frac{\text{d}P(t,T_1)}{P(t,T_1)}&=r_t\text{d}t+\sigma(t,T_1)\text{d}W_t^{(1)}
\\[6pt]
\frac{\text{d}P(t,T_2)}{P(t,T_2)}&=r_t\text{d}t+\sigma(t,T_2)\text{d}W_t^{(2)}
\\[6pt]
\text{d}W_t^{(1)}\text{d}W_t^{(2)}&=\rho\text{d}t
\end{align}$$
The $T_1$-forward measure $Q_{T_1}$ is defined such that all $T_1$-forward assets are martingales. Let us define:
$$\begin{align}
\tilde{B}_t &\triangleq \frac{B_t}{P(t,T_1)}
\\[6pt]
\tilde{P}(t,T_2)&\triangleq \frac{P(t,T_2)}{P(t,T_1)}
\end{align}$$
By Itô's Lemma, the forwards dynamics are:
$$\begin{align}
\frac{\text{d}\tilde{B}_t}{\tilde{B}_t} &= \sigma^2(t,T_1)\text{d}t-\sigma(t,T_1)\text{d}W_t^{(1)}
\\[6pt]
\frac{\text{d}\tilde{P}(t,T_2)}{\tilde{P}(t,T_2)} &=\left(\sigma^2(t,T_1)-\rho\sigma(t,T_1)\sigma(t,T_2)\right)\text{d}t+\Sigma(t)\cdot\text{d}W_t\end{align}$$
where:
$$\begin{align}
\Sigma(t)&\triangleq \bigg(-\sigma(t,T_1),\sigma(t,T_2)\bigg)
\\[2pt]
W_t&\triangleq \bigg(W_t^{(1)},W_t^{(2)}\bigg)
\end{align}$$
Using Girsanov theorem, we define the $T_1$-forward measure such that the following processes are Brownian Motions under $Q_{T_1}$:
$$\begin{align}
\tilde{W}_t^{(1)}&=W_t^{(1)}-\int_0^t\sigma(s,T_1)\text{d}s
\\[6pt]
\tilde{W}_t^{(2)}&=W_t^{(2)}-\rho\int_0^t\sigma(s,T_1)\text{d}s
\end{align}$$
which turn into martingales the forward dynamics:
$$\begin{align}
\frac{\text{d}\tilde{B}_t}{\tilde{B}_t} &= \sigma(t,T_1)\text{d}\tilde{W}_t^{(1)}
\\[6pt]
\frac{\text{d}\tilde{P}(t,T_2)}{\tilde{P}(t,T_2)} &=\Sigma(t)\cdot\text{d}\tilde{W}_t\end{align}$$
Therefore spot dynamics under the $T_1$-forward measure are:
$$\begin{align}
\frac{\text{d}B_t}{B_t}&=r_t\text{d}t
\\[6pt]
\frac{\text{d}P(t,T_1)}{P(t,T_1)}&=\left(r_t+\sigma^2(t,T_1)\right)\text{d}t+\sigma(t,T_1)\text{d}\tilde{W}_t^{(1)}
\\[6pt]
\frac{\text{d}P(t,T_2)}{P(t,T_2)}&=\left(r_t+\rho\sigma(t,T_1)\sigma(t,T_2)\right)\text{d}t+\sigma(t,T_2)\text{d}\tilde{W}_t^{(2)}
\end{align}$$
