How to obtain the counting of TTL pulses from an FPGA?

I'm currently working on a project in which I need to program an FPGA in order to connect it to a photon detector, that generates a TTL pulse when detecting a photon; I need to count this pulses, meaning, I need to count the photons being detected. The duration of each pulse according to the detector documentation is 20 ns.

I'm completely new to hardware programming, but I've been trying to keep up. So far, I was able to create the PulseCounter module and its testbench. The functionality I want is to set intervals in the scale of microseconds and then count the pulses from the detector during that time, up until a total time is completed, so I'm using frames of 100 μs during 1 second. Based on a paper I have, the number of photons is between 0 and 14 (it follows a Poisson distribution with a mean of 3.6) for a 254 μs time frame. Right now the pulse signal in the testbench is running continously. The counting seems to be working fine.

This is the module PulseCounter.v:

module PulseCounter (
    input wire clk,
    input wire reset,
    input wire photon_pulse,
    output reg [31:0] count,
    output reg interval_done
);

reg photon_pulse_dly; // Delayed version of photon_pulse
reg [31:0] timer;
reg [31:0] total_timer;

parameter INTERVAL = 5000; // 100 us = 0.0001 s @ 50MHz clock
parameter TOTAL_TIME = 50000000; // 1 s @ 50MHz clock

always @(posedge clk or posedge reset) begin
    if (reset) begin
        count <= 32'b0;
        timer <= 32'b0;
        total_timer <= 32'b0;
        photon_pulse_dly <= 0;
        interval_done <= 0;
    end else begin
        photon_pulse_dly <= photon_pulse;
        if (photon_pulse && !photon_pulse_dly) begin // Rising edge detected
            count <= count + 1;
        end
        if (timer == INTERVAL) begin
            timer <= 32'b0;
            interval_done <= 1;
            if (total_timer != TOTAL_TIME) begin
                total_timer <= total_timer + INTERVAL;
                count <= 32'b0; // Reset count for next interval
            end
        end else begin
            timer <= timer + 1;
            interval_done <= 0;
        end
    end
end

endmodule

And this is the testbench PulseCounter_tb.v:

`timescale 1ns / 1ps

module PulseCounter_tb;

    reg clk;
    reg reset;
    reg photon_pulse;
    wire [31:0] count;
    wire interval_done;

    // Instantiate the PulseCounter module
    PulseCounter uut (
        .clk(clk), 
        .reset(reset), 
        .photon_pulse(photon_pulse), 
        .count(count), 
        .interval_done(interval_done)
    );

    initial begin
        // Initialize inputs
        clk = 0;
        reset = 0;
        photon_pulse = 0;

        // Apply reset
        reset = 1;
        #10;
        reset = 0;        
    end
     
     initial begin
        // Generate a certain number of pulses
        repeat (5000) begin
            #20; photon_pulse = 1; // Pulse starts
            #20; photon_pulse = 0; // Pulse ends
        end
     
        // Stop the simulation after all pulses are generated
        #10000;
      $finish;
     end

    // Clock generator
    always #10 clk = ~clk;

endmodule

The FPGA I'm using is a DE10-Nano and I'm using Quartus. The next step would be retrieving this information and sending it to my PC. This is what I need help with. I've been looking and I found an alternative which is USB-to-UART connection. I don't know how to do this, and what would be the appropiate way to do it (whether the counting of each interval should be send altogether or at the end of the interval). I'm more familiar with high level programming languages, so the way I visualize it is to have a .txt with the counting for each interval, but then again, I'm lost here. I would appreciate a lot anyone's help.