Graphene and carbon fibers have a lot of implications, first of all a problem that some investigations suggest that graphene rib the celular membrane.
Other problem is that graphite to create graphene requires petroleum coke which is obtained heating the petoleum to up than 1400 °C no one living organism is able to resist that temperatures and obviosuly not able to produce that temperatures because proteins are denatured at even lower temperatures.
But here are the good points
Producing graphene from CO2, inspired in the ability of the plants to transform atmospheric CO2 to biomass using water, so the CO2 conversion to graphene is able using metalic enzymes and metalic plates based on copper and paladium, the current problem is the low efficiency with high energy required at temperatures of more than 1000 °C (notoriously less than the current coke process).
CO2 reduction to solid carbon alotropes, using liquid metalic electrocatalizers can transform the atmospherical CO2 to solid carbon, with a current low efficiency producing so few quantities.
Carbon nanofibers from CO2 using carbonate based electrolytes and iridium and niquel electrodes, apparently using a low energy got using a photovoltaic galvanic cell with a maximum of 2.5-2.7 volts and 2 amperes. This is a very positive option.
If you want there are bacterias synthesizing silicon composals https://pubs.rsc.org/en/content/articlelanding/2008/jm/b719528a#!divAbstract if you want to use things based of silicon like silicon carbide.
But someone with more chemistry knowledge can explain this better maybe I have misunderstood some things.
But this https://newatlas.com/bacteria-make-graphene/60529/ is the most positive way, bacterias synthesizing graphene with higher purity thatn chemical reduction process.
But you still having a big problem in the process:
heat petroleum or mineral carbon>>>get graphite>>>refine graphite>>>oxide graphite>>>reduce the oxide graphite>>get graphene>>>refine graphene
With the provived information you practically have solved the last three steps, so I would be focused on find how a living creature could produce graphite a lower temperature, but it is to be expected that, as another user mentioned, a biological process would be very different from an industrial process.