The work deals with molecular dynamics (MD) simulations of protonated, human telomeric i-motif deoxyribonucleic acid (DNA) with functionalized graphene. We studied three different graphene sheets: unmodified graphene with hydrogen atoms attached to their edges and two functionalized ones. The functionalization of graphene edge consists in attaching partially protonated or dissociated amine and carboxyl groups. We found that in all cases the protonated i-motif adsorbs strongly on the graphene surface. The biased MD simulations showed that the work necessary to drag the i-motif out from amine-doped graphene is about twice larger than that in other cases. In general, the system i-motif/amine-doped graphene stands out from the rest, e.g., in this case, the i-motif adsorbs its side with 3' and 5' ends oriented in the opposite to surface direction. In other cases, the DNA fragment is adsorbed to graphene by 3' and 5' ends. In all cases, the adsorption on graphene influences the i-motif internal structure by changing the distances between i-motif strands as well as stretching or shortening the DNA chain, but only in the case of amine-doped graphene the adsorption affects internal H-bonds formed between nucleotides inside the i-motif structure.