A morphism intertwining two induced representations

(1) is certainly false in general, but some things like it hold at least formally. See e.g. “Motivation” in Knapp (1986, §VII.3), which will send you to precise statements like Mackey’s Intertwining Number Theorem for representations induced from open subgroups (1951, Theorems 3 and 3’) (also found in Curtis-Reiner (1962, p. 327) for finite $G$), or Bruhat’s version for Lie groups, using distribution kernels (1956, pp. 160, 167, 171).

Edit: To answer your added comments, consider the case: $G$ is discrete, $D$ and $\Lambda$ are characters (1‑dimensional representations). Then the module $\mathrm{ind}_K^GD$ (where $G$ acts by $(g\varphi)(g')=\varphi(g^{-1}g')$) has the easily checked “reproducing kernel” property that its members satisfy $$ \varphi(g)=(gD^\bullet,\varphi) \tag i $$ where $(\cdot,\cdot)=$ inner product¹ and $D^\bullet\in\mathrm{ind}_K^GD$ is the function (cyclic vector) equal to $\overline D$ in $K$ and zero outside. Likewise for $\mathrm{ind}_Q^G\Lambda$, mutatis mutandis. So if $T\in\text{Hom}_G(\mathrm{ind}_K^GD,\mathrm{ind}_Q^G\Lambda)$ and we let $t:=T^*\Lambda^\bullet$ (star $=$ adjoint), we get (except for a complex conjugate in the notation) your desired result (1): $$ (T\varphi)(g) =(g\Lambda^\bullet,T\varphi) %=(gT^*\Lambda^\bullet,\varphi) =(gt,\varphi) =\sum_{g'K\in G/K}\overline t(g^{-1}g')\varphi(g'). \tag{ii} $$ Next one observes that (ii) applied to $\varphi=D^\bullet$ gives $TD^\bullet=t^\vee$ where $f^\vee(g):=\overline f(g^{-1})$. So $t$ is in $(\mathrm{ind}_K^GD)\cap(\mathrm{ind}_Q^G\Lambda)^\vee$, hence it satisfies the relation (complex conjugate of yours) $$ t(q^{-1}gk) = D(k^{-1})t(g)\Lambda(q). \tag{iii} $$ Such a function is determined by one value per double coset $QgK$. Moreover, as one sees by putting $q=gkg^{-1}$ in (iii), this value must vanish unless $$ k\mapsto\Lambda(gkg^{-1}) \style{font-family:sans-serif}{\text{ coincides with }} D \style{font-family:sans-serif}{\text{ on }} K\cap g^{-1}Qg. \tag{iv} $$ (This is the “intertwining two representations defined on a subgroup” you ask about.) Summing up, we get that $\dim(\text{Hom}_G(\mathrm{ind}_K^GD,\mathrm{ind}_Q^G\Lambda))\leqslant \sharp\{$double cosets $QgK\in Q\backslash G/K$ satisfying (iv)$\}$, which is the intertwining number theorem. It goes back to Shoda (1933, p. 251); adapting it to non-discrete groups (with sums replaced by integrals, kernels by distributions, etc.) raises hard analytic problems famously solved in some cases by Knapp, Kunze, Stein, etc.

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^{1. $(\varphi_1,\varphi_2):=\sum_{gK\in G/K}\overline{\varphi_1(g)}\varphi_2(g)$, where bar $=$ complex conjugate and there is one term per coset (within which choice of $g$ doesn’t matter, thanks to the equivariance condition).}