The answer is no, which you can see because plenty of counterexamples exist. For example, a main-sequence star with spectral type B burns hydrogen and has an effective temperature of 10,000-30,000 K. A type-B star with an effective temperature of 15,000 K has a peak blackbody wavelength of 199 nm, which is nowhere close to any hydrogen spectral lines. The fact that the Sun's effective temperature is kind of (but not particularly) close to a particular hydrogen line is a coincidence, and, given that there are many hydrogen lines from the UV to the IR range, not a particularly rare one, either.
As to why your deduction isn't true, think of it this way: the effective temperature of a star is the temperature of the outer surface of the star. Fusion reactions do not occur at the surface of the star, but rather in the core, where the temperature is much higher (the specifics here depend on several variables, as the dynamics within a star depend heavily on its size and the presence of heavier elements). As such, there is no reason to expect any sort of direct relation between the temperature of the surface and the element being burned. In fact, for most stars on the main-sequence, which burn primarily hydrogen, you can find either a red giant or a white dwarf (or both!) with the same effective temperature; red giants primarily burn heavier elements, and white dwarfs don't burn anything at all. So it should be clear that the effective temperature alone can't tell you anything about the nuclear reactions taking place.
In addition, the spectral lines present in the stellar atmosphere also aren't really related to the nuclear reactions taking place. These absorption lines come about because electrons attached to atoms in a certain state are excited by radiation of a specific frequency into a higher energy state. This requires that atoms with electrons still attached in lower energy states exist (bare ions do not produce spectral lines), and as such, occurs at energies/temperatures far lower than the temperatures required by/energy released by nuclear fusion. The intensity of a spectral line in a stellar atmosphere at a given temperature tells you about the composition of the outer layer, which is not always influenced by the fuel being burned. For example, large stars have a radiative outer layer, which means that convection to the surface doesn't really occur and the fusion products tend to stay in the core, not really influencing the composition of the outer layer. Midrange stars have a radiative core and a convective outer layer, so there is only limited transport between the core and the surface. Only in the smallest stars does convection from the core to the surface occur.