Is it NP complete to decide whether a graph has $k$ disjoint triangles?
For the special case where $n$ is a multiple of 3 and $k=n/3$, this is the PARTITION-INTO-TRIANGLES problem, which is proved to be NP-complete here.
The idea is to reduce 3-DIMENSIONAL-MATCHING (3DM) to PARTITION-INTO-TRIANGLES. I quote the construction here.

                                           
Let $X = \{x_1, x_2, \ldots, x_q\}$, $Y = \{y_1, y_2, \ldots, y_q\}$, $W = \{w_1, w_2, \ldots, w_q\}$ and $M \subseteq X\times Y\times Z$ be the generic input instance of 3DM. Construct an initial set of vertices $V$, called the &quot;public vertices&quot;, as the union of $X$, $Y$ and $W$. For every triple $m = (x_a, y_b, w_c)$ in $M$, construct the graph $G_m = (V_m, E_m)$, depicted in Figure 1. Here $V_m$ contains the three public vertices $x_a$, $y_b$ and $w_c$ plus three &quot;private vertices&quot; $v_m^1$, $v_m^2$ and $v_m^3$. The ten edges in $E_m$ are $v_m^1v_m^2$, $v_m^1v_m^3$, $v_m^2v_m^3$, $x_av_m^3$, $x_av_m^2$, $y_bv_m^3$, $y_bv_m^1$, $w_cv_m^1$, $w_cv_m^2$ and $x_1y_b$. The whole graph $G_M$ is obtained from the union of all the gadgets $G_m$ as follows: $G_M = (V_M, E_M) = (\cup_m V_m,\cup_m E_m)$.