First, note that the posted code will result in an index-out-of-bounds error, because after Num_of_SNPs iterations of the main loop your j value will be ncol(geno.data)-1 and you're accessing columns 2*j-1 and 2*j. I'm assuming you instead want columns 2*count-1 and 2*count and j can be removed.
Vectorization is extremely important for writing fast R code. In your code you're calling the paste function 313 times, each time passing vectors of length 1. It's much faster in R to call paste once passing vectors of length 313. Here are the original and vectorized interiors of the main for loop:
# Original
get.pval1 <- function(count) {
  for (i in 1:nrow(geno.data)){
    alleles[i]<- levels(genotype(paste(geno.data[i,c(2*count -1, 2*count)], collapse = "/")))
  }
  g2 <- genotype(alleles)
  HWE.chisq(g2)[3]
}
# Vectorized
get.pval2 <- function(count) {
  g2 <- genotype(paste0(geno.data[,2*count-1], "/", geno.data[,2*count]))
  HWE.chisq(g2)[3]
}
We get about a 20x speedup from the vectorization:
library(microbenchmark)
all.equal(get.pval1(1), get.pval2(1))
# [1] TRUE
microbenchmark(get.pval1(1), get.pval2(1))
# Unit: milliseconds
#          expr       min        lq      mean    median        uq       max neval
#  get.pval1(1) 299.24079 304.37386 323.28321 307.78947 313.97311 482.32384   100
#  get.pval2(1)  14.23288  14.64717  15.80856  15.11013  16.38012  36.04724   100
With the vectorized code, your code should finish in about 177616*.01580856 = 2807.853 seconds, or about 45 minutes (compared to 16 hours for the original code). If this is still not fast enough for you, then I would encourage you to look at the parallel package in R. The mcmapply should give a good speedup for you, since each iteration of the outer for loop is independent.