Spaces was designed to be inter-operable with the S3 API (for more details see here). This includes the ability to presign URLs. The only difference when porting from a code base originally targeting S3 will be configuring the correct endpoint (e.g. $REGION.digitaloceanspaces.com).

Here's a quick example with vanilla PHP:

<?php

// Included aws/aws-sdk-php via Composer's autoloader
// Installed with: composer.phar require aws/aws-sdk-php
require 'vendor/autoload.php';
use Aws\S3\S3Client;

// Configure a client using Spaces
$client = new Aws\S3\S3Client([
        'version' => 'latest',
        'region'  => 'nyc3',
        'endpoint' => 'https://nyc3.digitaloceanspaces.com',
        'credentials' => [
                'key'    => 'MY_SPACES_KEY',
                'secret' => 'MY_SPACES_SECRET,
            ],
]);

$cmd = $client->getCommand('GetObject', [
    'Bucket' => 'examplebucket',
    'Key'    => 'file.ext'
]);

$request = $client->createPresignedRequest($cmd, '+5 minutes');
$presignedUrl = (string) $request->getUri();

echo $presignedUrl;