Assuming that your model of the dynamics is correct (i.e. $F_k$, $B_k$ and $H_k$) then the good news is that any (positive semi-definite) choice for $Q$ and $R$ will result in a state estimator that is stable, so the estimated state converges to the true state in the absence of disturbances $w_k$ and $v_k$ and converge to some bound around the true state when there are disturbances.
If the covariance matrices are off by some scalar factor $\alpha$, then one still yields the same state estimate. This can be seen by looking at the matrix update equations from the Kalman filter
\begin{aligned}
\mathbf {P} _{k\mid k-1}&=\mathbf {F} _{k}\mathbf {P} _{k-1\mid k-1}\mathbf {F} _{k}^{\textsf {T}}+\mathbf {Q} _{k},\\
\mathbf {S} _{k}&=\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}+\mathbf {R} _{k},\\
\mathbf {K} _{k}&=\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {S} _{k}^{-1},\\
\mathbf {P} _{k|k}&=\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k|k-1}.
\end{aligned}
Namely, instead of $\mathbf {P} _{k|i}$, $\mathbf {Q} _{k}$ and $\mathbf {R} _{k}$ using $\mathbf {P'} _{k|i}=\alpha\mathbf {P} _{k|i}$, $\mathbf {Q'} _{k}=\alpha\mathbf {Q} _{k}$ and $\mathbf {R'} _{k}=\alpha\mathbf {R} _{k}$ respectively yields
\begin{aligned}
\mathbf {P'} _{k\mid k-1}&=\mathbf {F} _{k}\mathbf {P'} _{k-1\mid k-1}\mathbf {F} _{k}^{\textsf {T}}+\mathbf {Q'} _{k} = \alpha\left(\mathbf {F} _{k}\mathbf {P} _{k-1\mid k-1}\mathbf {F} _{k}^{\textsf {T}}+\mathbf {Q} _{k}\right),\\
\mathbf {S'} _{k}&=\mathbf {H} _{k}\mathbf {P'} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}+\mathbf {R'} _{k} = \alpha\left(\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}+\mathbf {R} _{k}\right),\\
\mathbf {K'} _{k}&=\mathbf {P'} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {S'} _{k}^{-1} = \frac\alpha\alpha\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {S} _{k}^{-1} = \mathbf {K} _{k},\\
\mathbf {P'} _{k|k}&=\left(\mathbf {I} -\mathbf {K'} _{k}\mathbf {H} _{k}\right)\mathbf {P'} _{k|k-1} = \alpha\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k|k-1}.
\end{aligned}
So the gain matrix $\mathbf {K'} _{k}$ remains the same under any scalar factor $\alpha$, so the state update step will also remain the same. One might run into some numerical issues when $\alpha$ is really big or really small, but on the other side $\alpha$ can also be used for normalization and make the Kalman filter more numerically robust. Such normalization will also change the direct relationship between $\mathbf {P'} _{k|k}$ and the uncertainty of the state estimate, so keep the factor $\alpha$ in mind when interpreting $\mathbf {P'} _{k|k}$.
I can not answer your more general question of the exact impact of using covariance matrices that deviate is more ways that just a scalar factor. It can be noted that chosing the $Q$ and $R$ matrices is often also considered a tuning step of making a Kalman filter. These matrices can also be used to reduce the impact of other types of disturbances (so other than zero mean Gaussian white noise).
Best Answer
I find the equation in the screenshot very odd, and I work with Kalman filters quite often. Don't get too caught up on that particular equation.
The important takeaway is that the derivation of the Kalman "gain" matrix $K_k$ is based on minimizing the trace of $P_k$. The diagonal elements of $P_k$ (that are summed together in the trace) are the variances of the estimation errors for the individual elements of the state vector. For example, if element $i$ of the state vector $x_k$ was a position in meters, then entry $(P_k)_{ii}$ is the variance of the estimation error for state element $i$ (in units m$^2$). And it turns out that minimizing the sum of the estimation error variances over all the elements of the state is a good criterion for obtaining good estimates.