1. Use the reshape2 package to convert that matrix to a format with the following columns: Gene (these will repeat once per sample), number of mutations, and sample.
2. Use ggplot to make the dot plot. If you set colour= to the sample column then it'll generate the legend for you.

This is a very direct question to get others to do your homework but I am assuming you have absolutely no background in R because I see no code showing that you at least tried. From hereon, you should work on your R skills. Meanwhile you can use this code:

library(ggplot2)
library(reshape2)

# use melt function to transform your dataset for ggplot2
# it melts your dataset such that first column contains genes, second column has sample names and third column has the mutation value
dat = melt(matrix)
head(dat)

   Var1  Var2 value
1 Q0010 D784G     0
2 Q0032 D784G     0
3 Q0055 D784G     1
4 Q0075 D784G     0
5 Q0080 D784G     0
6 Q0081 D784G    15

# create plot using ggplot function. geom_point() is used for creating dot plots
ggplot(data = dat,aes(x = Var1, y = value)) + 
  geom_point(data = dat,aes(pch = factor(Var2), colour = factor(Var2))) + 
  xlab('Genes') + scale_shape_discrete(name="LegendTitle") + scale_colour_discrete(name="LegendTitle")

Play with the ggplot2 package to understand what each function is doing.