Financial inclusion in most studies is linked to growth and poverty reduction. However, one can argues that the move of money from boxes and under mattresses to the financial sector (precisely banks), could pose a challenge to central bank regarding the quantity of money supplied. Central bank could supply a quantity of money which becomes higher than what the economy needs due to the new flow of money to banks (which was not expected by the CB).. This could create inflation (indirect relationship).
The second option could be summarized as follows: as financial inclusion lead to financial development (expel, increase of credit to the private sector) hence you could then link the financial development to inflation and there are numerous papers studying this issues (see the various papers by Asli Demirguc-Kunt, in the following link: https://ideas.repec.org/f/pde226.html)
FINANCIAL INCLUSION: Making sure that the low income segment of society has access to financial institution and services with the aim of accessibility and cost reduction. There are 4 requirements in order for financial inclusion to succeed: (i) accessibility; (ii) safe financial institution, (iii) institution continuity or operation, and (iv) competition within the financial market in order to drive down costs.
INFLATION: By definition inflation is the change in price level. The link between financial inclusion and inflation would be indirect if any. One could argue that if the poor have access to the bank they would put money in the bank and would not spend it, therefore, curve inflation---however, if the poor are poor to begin with, i.e. having no money, accessing the bank could not be meaningful. If accessing the financial institution means accessing financial services, i.e. the poor who do not have enough cash to buy a house may access mortgage and make affordable monthly payment over 30 years---in this case, the effect on changes in price level (inflation) may be direct. However, this is speculative. Use the actual data for both financial inclusion and inflation year-by-year to verify the relationship and test for the statistical significance of the relationship.
The World Bank maintains an index for global financial inclusion. See here:
MEASUREMENT: You could run the relationship test between financial inclusion by taking data of financial inclusion country-by-country in time series year-by-year and run it against inflation data from the IMF's Global Economic Outlook Report. I attached the IMF data here. For financial inclusion, Work Bank site.
Inflation is positively correlated with financial inclusion. This is intuitively appealing as the increased circulation of money as well as the creation of additional demand would lead people to use financial services more often.