How do you convert a Unix timestamp to Jalali/Shamsi/Persian format?

you can use toLocalDateString()

let mydate = new Date(timestamp);
let mypersiandate = mydate.toLocaleDateString('fa-IR');
console.log(mypersiandate);

I would suggest using moment.js (https://momentjs.com/) which is reliable JavaScript Time library that allows you to format your timestamp in JavaScript. Below is an example of how you can parse a timestamp and format it to whatever you want using it.

//formatting Unix timestamp.
var date = moment.unix(value).format("MM/DD/YYYY");

You also tagged localization which can be done by using;

 var localeDate = moment(date).locale("LT");

More examples can be found on there website.

This in conjunction with https://www.npmjs.com/package/jalali-date will get you your jalali date.

There is a moment.js extension for Persian here also https://www.npmjs.com/package/moment-jalaali (From moment to Jalali)

Another Jalali conversion library https://www.npmjs.com/package/jalaali-js (To Jalai)

An example fiddle using moment.js Jalali conversion from Unix Timestamp https://jsfiddle.net/uw82ozpd/9/

Relevant code snippet with comments:

var UnixValue;
var JalaliValue;

$(document).ready(function() {

//get the Unix Date from HTML
var UnixValue = $(".Unix-Date").html();

//Get a moment timestamp in the format simmilar to our next conversion
var date = moment.unix(UnixValue).format("MM/DD/YY");

//Convert from normal moment to our jalali moment exstension using j's as below
var JalaliValue = moment(date).format('jYYYY/jM/jD');

$(".Jalali-Date").text(JalaliValue);

});

I have these simple functions:

function convertTimeStampToJalali(timestamp){
    var date = new Date(timestamp);
    if(!date)
        return false;
    return ( gregorian_to_jalali(date.getFullYear(),(date.getMonth()+1),date.getDate()) );
}//end of function convertTimeStampToJalali

function gregorian_to_jalali(gy,gm,gd){
    g_d_m=[0,31,59,90,120,151,181,212,243,273,304,334];
    if(gy > 1600){
        jy=979;
        gy-=1600;
    }else{
        jy=0;
        gy-=621;
    }
    gy2=(gm > 2)?(gy+1):gy;
    days=(365*gy) +(parseInt((gy2+3)/4)) -(parseInt((gy2+99)/100)) +(parseInt((gy2+399)/400)) -80 +gd +g_d_m[gm-1];
    jy+=33*(parseInt(days/12053)); 
    days%=12053;
    jy+=4*(parseInt(days/1461));
    days%=1461;
    if(days > 365){
        jy+=parseInt((days-1)/365);
        days=(days-1)%365;
    }
    jm=(days < 186)?1+parseInt(days/31):7+parseInt((days-186)/30);
    jd=1+((days < 186)?(days%31):((days-186)%30));
    return [jy,jm,jd];
}//end of function gregorian_to_jalali