Is a fusion process possible inside an accretion disk? Is a fusion process possible inside an accretion disk? As the speed of matter inside an accretion disk approaches the speed of light can an incoming object collide with the disk material and cause a nuclear process?
 A: Ion temperatures inside hot accretion disks (so called advection–dominated accretion flows) could reach $10^{12}\,\text{K}$, hot enough to produce nuclear reactions, including fusion. Note, that this is the temperature of random ion motion relative to the average accretion disk flow, there is no need for additional “incoming object” to initiate nuclear reactions.
The overall power produced from nuclear reactions in accretion disks is expected to contribute less than 1% to overall power radiated which is in turn about 5-10% of total mass-energy of accreted matter, so the role of nuclear reactions in the dynamics of accretion disks is not that significant.
For completeness let us also mention a concept of “black hole as a particle accelerator”: even without accretion disk to heat up plasma, two particles falling into rotating (Kerr) black hole from (almost) zero velocity far away, can collide near (but still outside) the horizon with energy in the center of mass frame being arbitrarily high. So collisions near a black hole could be the source of most energetic reactions, potentially far exceeding those being probed at the LHC, let alone nuclear fusion. And some of the reaction products have a chance to escape the near horizon region to be observed (though more likely is that most reaction products would fall into the black hole).
References: For general overview of accretion disks astrophysics see the book/lecture notes:

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*Romero, G. E., & Vila, G. S. (2013). Introduction to black hole astrophysics (Vol. 876). Springer, free pdf of lecture notes.

For an overview of “black holes as a particle accelerators” aimed at students of general relativity:

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*Harada, T., & Kimura, M. (2014). Black holes as particle accelerators: a brief review. Classical and Quantum Gravity, 31(24), 243001, doi:10.1088/0264-9381/31/24/243001, arXiv:1409.7502.

