Herein, we present results from observations of the 12CO (J = 1–0), 13CO (J = 1–0), and 12CO (J = 2–1) emission lines toward the Carina nebula complex (CNC) obtained with the Mopra and NANTEN2 telescopes. We focused on massive-star-forming regions associated with the CNC including the three star clusters Tr 14, Tr 15, and Tr 16, and the isolated WR-star HD 92740. We found that the molecular clouds in the CNC are separated into mainly four clouds at velocities −27, −20, −14, and −8 km s−1. Their masses are 0.7 × 104 M$\odot$, 5.0 × 104 M$\odot$, 1.6 × 104 M$\odot$, and 0.7 × 104 M$\odot$, respectively. Most are likely associated with the star clusters, because of their high 12CO (J = 2–1)/12CO (J = 1–0) intensity ratios and their correspondence to the Spitzer 8 μm distributions. In addition, these clouds show the observational signatures of cloud–cloud collisions. In particular, there is a V-shaped structure in the position–velocity diagram and a complementary spatial distribution between the −20 km s−1 cloud and the −14 km s−1 cloud. Furthermore, we found that SiO emission, which is a tracer of a shocked molecular gas, is enhanced between the colliding clouds by using ALMA archive data. Based on these observational signatures, we propose a scenario wherein the formation of massive stars in the clusters was triggered by a collision between the two clouds. By using the path length of the collision and the assumed velocity separation, we estimate the timescale of the collision to be ∼1 Myr. This is comparable to the ages of the clusters estimated in previous studies.