Thank you for the link to the doc.
The first method described in the doc (doveadm quota) does not work, it throws
an error:

   "Fatal: Unknown command 'quota', but plugin quota exists. Try to set mail_plugins=quota"

with the 2nd method (delete records from the database table used_quota) is more successful.

However, now it shows the used_quota as zero, until the user logs into his e-mail. That's not ideal either - I cannot control when they will log in, and so it reports a false figure of zero for maybe a long time.

I would like to have a simple way that actually shows the present quota usage. The "doveadm quota" seems to be better, but I don't find a way on how to make it work.In dovecot.conf there is already lines with "mail_plugins = quota". If anything is known on that already please let me know.

It's not about how I use these records, it is how the mail system, and the Thunderbird clients, use this.
When a user is above quota (and that "above quota" decision comes from that figure in the used_quota table), then the user cannot send and receive mail. And it is a complete pain to get a user back down - you can literally delete ALL the mails on a Thunderbird client, and still get "mailbox is 100% full". No amount of restart and compacting folders help.
So the solution is to increase the quota of the user ... which soon ends up everybody having monstruous quotas. When I now used your suggestion above from the doc (deleting the records in used_quota table) the quota percentages dropped from 95% down to 10% for many users.

That's the background. I am not saying it's a bug in iRedMail, I may have outdated Thunderbird clients, too, etc, I I just ask for data if that is a known problem in the dovecot quota system.

Regarding the dovecot config - it is the default config from iRedMail.

Here is the output of dovecot -n

# 2.0.19: /etc/dovecot/dovecot.conf
# OS: Linux 3.2.0-38-generic-pae i686 Ubuntu 12.04.4 LTS
auth_mechanisms = PLAIN LOGIN
default_process_limit = 1024
dict {
  acl = mysql:/etc/dovecot/dovecot-share-folder.conf
  expire = db:/var/lib/dovecot/expire/expire.db
  quotadict = mysql:/etc/dovecot/dovecot-used-quota.conf
}
disable_plaintext_auth = no
first_valid_uid = 1001
last_valid_uid = 1001
listen = *
log_path = /var/log/dovecot.log
mail_gid = 1001
mail_location = maildir:/%Lh/Maildir/:INDEX=/%Lh/Maildir/
mail_uid = 1001
managesieve_notify_capability = mailto
managesieve_sieve_capability = fileinto reject envelope encoded-character vacation subaddress comparator-i;ascii-numeric relational regex imap4flags copy include variables body enotify environment mailbox date ihave
namespace {
  inbox = yes
  location =
  prefix =
  separator = /
  type = private
}
namespace {
  list = children
  location = maildir:/%%Lh/Maildir/:INDEX=/%%Lh/Maildir/Shared/%%u
  prefix = Shared/%%u/
  separator = /
  subscriptions = yes
  type = shared
}
passdb {
  args = /etc/dovecot/dovecot-ldap.conf
  driver = ldap
}
plugin {
  acl = vfile
  acl_shared_dict = proxy::acl
  auth_socket_path = /var/run/dovecot/auth-master
  autocreate = INBOX
  autocreate2 = Sent
  autocreate3 = Trash
  autocreate4 = Drafts
  autocreate5 = Junk
  autosubscribe = INBOX
  autosubscribe2 = Sent
  autosubscribe3 = Trash
  autosubscribe4 = Drafts
  autosubscribe5 = Junk
  expire = Trash 7 Trash/* 7 Junk 30
  expire_dict = proxy::expire
  quota = dict:user::proxy::quotadict
  quota_rule = *:storage=1G
  quota_warning = storage=85%% quota-warning 85 %u
  quota_warning2 = storage=90%% quota-warning 90 %u
  quota_warning3 = storage=95%% quota-warning 95 %u
  sieve = /%Lh/sieve/dovecot.sieve
  sieve_dir = /%Lh/sieve
  sieve_global_dir = /maildisk/sieve
  sieve_global_path = /maildisk/sieve/dovecot.sieve
}
protocols = pop3 imap sieve
service auth {
  unix_listener /var/spool/postfix/dovecot-auth {
    group = postfix
    mode = 0666
    user = postfix
  }
  unix_listener auth-master {
    group = vmail
    mode = 0666
    user = vmail
  }
  unix_listener auth-userdb {
    group = vmail
    mode = 0660
    user = vmail
  }
}
service dict {
  unix_listener dict {
    group = vmail
    mode = 0660
    user = vmail
  }
}
service imap-login {
  process_limit = 1024
  service_count = 1
}
service pop3-login {
  process_limit = 1024
  service_count = 1
}
service quota-warning {
  executable = script /usr/local/bin/dovecot-quota-warning.sh
  unix_listener quota-warning {
    group = vmail
    mode = 0660
    user = vmail
  }
}
ssl_cert = </etc/ssl/certs/iRedMail_CA.pem
ssl_key = </etc/ssl/private/iRedMail.key
userdb {
  args = /etc/dovecot/dovecot-ldap.conf
  driver = ldap
}
protocol lda {
  auth_socket_path = /var/run/dovecot/auth-master
  lda_mailbox_autocreate = yes
  log_path = /var/log/sieve.log
  mail_plugins = quota sieve autocreate
  postmaster_address = root
}
protocol imap {
  imap_client_workarounds = tb-extra-mailbox-sep
  mail_max_userip_connections = 100
  mail_plugins = quota imap_quota autocreate
}
protocol pop3 {
  mail_max_userip_connections = 100
  mail_plugins = quota
  pop3_client_workarounds = outlook-no-nuls oe-ns-eoh
  pop3_uidl_format = %08Xu%08Xv
}