The problem is that you are not checking for a space character. Check it as follows:

if (a1[i] >= 'A' && a1[i] <= 'Z' || a1[i] == ' ')

Another problem with your code is changing the value of count to 1 and 2 in each iteration whereas it should be changed when the loop terminates. Given below is the corrected code:

import java.util.Scanner;

public class Main {
    public static void main(String[] args) {
        int count = 0, i;
        Scanner sc = new Scanner(System.in);
        System.out.println("Inmate's name:"); 
        String name = sc.nextLine();
        System.out.println("Inmate's father's name:");
        String fname = sc.nextLine();

        String s3 = name.toUpperCase();
        String s4 = fname.toUpperCase();

        char[] a1 = s3.toCharArray();
        char[] a2 = s4.toCharArray();
        for (i = 0; i < a1.length; i++) {
            if (!(a1[i] >= 'A' && a1[i] <= 'Z' || a1[i] == ' ')) {
                System.out.print("Invalid name1");
                count = 0;
                break;
            }
        }

        // If 'i' reached a1.length, it means no invalid character was found
        if (i == a1.length) {
            count = 1;
        }

        if (count == 1) {
            for (i = 0; i < a2.length; i++) {
                if (!(a2[i] >= 'A' && a2[i] <= 'Z' || a2[i] == ' ')) {
                    System.out.print("Invalid name");
                    break;
                }
            }

            // If 'i' reached a2.length, it means no invalid character was found
            if (i == a2.length) {
                count = 2;
            }
        }

        if (count == 2) {
            System.out.print(s3 + " " + s4);
        }
    }
}

Additional note:

You can make your code much shorter by using regex as follows:

import java.util.Scanner;

public class Main {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        System.out.print("Inmate's name: ");
        String name = sc.nextLine();
        System.out.print("Inmate's father's name: ");
        String fname = sc.nextLine();
        if (name.matches("[A-Za-z\\s]+") && fname.matches(("[A-Za-z\\s]+"))) {
            System.out.println(name.toUpperCase() + " " + fname.toUpperCase());
        } else if (!name.matches("[A-Za-z\\s]+")) {
            System.out.println("Inmate's name is invalid");
        } else if (!fname.matches(("[A-Za-z\\s]+"))) {
            System.out.println("Inmate's father's name is invalid");
        }
    }
}

The explanation of the regex, [A-Za-z\\s]+:

1. A-Za-z is for alphabets.
2. \\s is for space.
3. The + at the end of [A-Za-z\\s]+ means more than one occurrences are allowed.

A sample run:

Inmate's name: Ram Kumar
Inmate's father's name: Raj Kumar
RAM KUMAR RAJ KUMAR

Another sample run:

Inmate's name: Ram5 Kumar
Inmate's father's name: Raj Kumar
Inmate's name is invalid

Another sample run:

Inmate's name: Ram Kumar
Inmate's father's name: Raj5 Kumar
Inmate's father's name is invalid