We perform calculations using the semiclassical Fourier approach for quasimolecular spectra induced by thermal collisions of the excited calcium atom (Ca*) with the helium atom in the ground state (He) for the following quasimolecular optical transitions: (Ω = 0⁺, 4s4p ¹P) ↔ (Ω = 0⁺, 4s^{2 1}S₀) and (Ω = 0⁺, 4s3d ¹D) ↔ (Ω = 0⁺, 4s^{2 1}S₀) in the Ca-He quasimolecule. As a comparison with the experimental results obtained under gas-cell conditions, we average the spectra under study over the impact parameters and the Maxwellian distribution of the colliding particles. For the asymptotically forbidden Ca(4s^{2 1}S₀ → 4s3d ¹D₂)-He quasimolecular transition, the first comparison between the calculated spectral line profile and the experimental one is made not only in shape, but also in absolute value. We clarify the physical mechanisms for the initiation of a spectral line satellite at the position of the forbidden atomic transition being investigated. The reasonable agreement confirms that the spectral lineshapes of the asymptotically forbidden transitions are formed due to the interplay between the structure of the potential energy curves involved and the strong dependence of the radiative width.