Yes, there are a bunch of methods other than elbow method which you can use instead. One of the most prominent of this is Silhouette method or average Silhouette method which basically try to find out how similar are the points or elements in a cluster are to their nearest points across all clusters whereas we calculate distance from each point of the cluster from it's center in elbow method.
Suppose, you take a set of clustering options ranging from 0 to 20. Just like elbow method for each clustering option, the method will generate silhouette score. Finally, the clustering option with the highest average silhouette score will be selected.
Hi Minakshi Kaushik , I personally work a lot with this short tutorial on cluster analyses, which introduces the silhouette method already mentioned by Mohd. Farhan Israk Soumik , the GAP statistic, and the consensus-based alorithm alongside the elbow method and already comes with R code for each method's implementation: