Are there any scenarios in which two stars could collide, and simply fuse without triggering a supernova. Maybe a mid-sized star slowly spiraling in together with a smaller star? It seems like any time stars collide it's incredibly violent. It seems that one star always gets torn to pieces and accretes matter onto the other, or ends in a red nova. Is there any way this process could be calmer?
The merger of two stars may only result in a supernova if the merging stars are white dwarfs, or possibly a white dwarf with a neutron star, and even in these scenarios a supernova is not certain.
Other mergers do not cause supernovae. Examples are the formation of contact binaries (W UMa binaries), which may then fully merge to produce fast-rotating FK Com stars, or more generally, the merger of main sequence stars is thought to produce blue stragglers. It is possible that some of these events may result in some sort of outburst, but nothing like a supernova - e.g Tylenda et al. (2011).
Are there any scenarios in which two stars could collide, and simply fuse without triggering a supernova
As said in the comments, binary evolution of stars is very uncertain, but there exist numerous codes (i.e. MESA, STARTRACK, BSE, etc...) which incorporate LOTS of physical processes over the course of millions of years (and if they evolve to black holes billions of years) of binary evolution. Ono Pols has a nice textbook free on his website here.
Stellar binaries usually experience at least a couple mass-transfer (MT) phases after the binary components fill their Roche lobes. To put it simply, MT is either stable or it is unstable, but either way it results in the donor star losing it's envelope so that it's core emerges (which will later turn into a compact object). In stable MT, the donor star transfers some mass to its companion and thusly loses rotational angular momentum (it's spin decreases) while the accretor's spin increases. In unstable mass transfer, the mass transfer rate is too much for the accretor to handle so that the binary is engulfed by the envelope, this is also called a Common Envelope (CE). The CE results in either 1) the envelope is ejected from the binary resulting in a drastic decrease in the binary separation, or 2) the binary merges leaving a single star which accretes the leftover envelope.
So, I think the case 2) is an answer to your question. Keep in mind that CE is very very uncertain, as Postnov explains, and it is a relatively quick process (less than 1 million years). It's worth noting that case 1) of CE is crucial for forming compact binaries that have small separations so that they merge within the lifetime of the universe (so we can measure they're gravitational waves).
For your nutrition, here in Postnov's review on compact binary evolution the section on CE is lovely (section 3.6 on page 35).
Of course, if you're willing to include black hole binaries in your question, then their merging due to the emission of gravitational waves is certainly a calmer scenario.