In R programming you can simply use the built-in vector that the RStudio provides for abbreviated months, month.abb and then after that just perform the indexing on these with the sequence of your x-axis tick labels, using modulus 12, %% 12. First, turn the x-axis off (xaxt="n").

plot(NULL,xlim=c(0,30), ylim=c(0,350), xlab="months", ylab="values in mm", main="procedure", xaxt="n")

And then create the desired labels.

(xlabs <- paste(month.abb[1+seq(0,30,5) %% 12])

# [1] "Jan" "Jun" "Nov" "Apr" "Sep" "Feb" "Jul"

The above labels only have the months. If you want to add the years, then that's also possible, perhaps underneath if space is an issue. For example, using paste0.

xlabs <- paste0(xlabs, "\n", c(rep(2018,3), rep(2019,2), rep(2020,2)))

axis(side=1, at=seq(0,30,5), labels=xlabs, cex.axis=0.8)